text:
Composing free and open online educational resources

Introduction
Free and open educational resources have become one of the most discussed topics in the field of education. Projects such as MIT Open courseware, Open Access, Wikipedia, Wikibooks and Wikimedia Commons have challenged traditional methods of delivering education resources and also the methods of creating them.

The free software movements idea of developing free, libre and open source software, as well as the Creative Commons search for alternatives to traditional copyright, have had an everlasting effect on the ways we think about education and educational resources.

The course readings and the assignments in this course will familiarize participants with the main concepts related to open education resources and to the historical and philosophical ideas behind them. The participants will also do their own projects where they will learn to create and participate in projects producing free and open educational resources.

Course / Class News
The news of the course are posted to the course blog by the facilitators:

 * http://oercourse.wordpress.com/

This course is targeted for teachers and teacher-students who do not have prior knowledge or skills related to free and open education resources.

Being a student or teacher-student, however, is not a requirement for participation. Motivation and interest on the topic are the basic requirements.

Language Policy
Because the idea of this project is that the participants can help each other in their course work we need a single common language when working with the assignments (see below). This language is English. However, you can use other languages when you create your own educational resources in the project work for this course (more information about projects below).

Participants
You may preliminary register / announce your interest to take part in the course starting in January. Please, add your name to the page:

 * Composing free and open online educational resources/Participants/Next start

The estimated workload for the participants is 4-6 hours of study work per week. Students are expected to spend this time to read, watch and listen online materials and to complete the assignments according to the schedule.

Objectives
During the course participants will become familiar with the idea of co-creation of learning resources. After the course participants will have the skills to find, create and share free and open education resources online. People will be familiar with the concepts of:

 * free and open culture
 * free, libre, open educational resources
 * commons based peer-production
 * educational resource design
 * online study project
 * media production

As part of the course assignment participants will be expected to create a free and open education resource. During the projects participants will learn to take digital images, to record audio and video and to use these media as educational resources.

Class Meetings
This class meets mainly asynchronously online in the blogs of the participants and in a mailing list. The participants are recommended to hang-out in the #lemill IRC channel (Freenode) for synchronous chat (a web client is available).

Assignments

 * The course syllabus, with the weekly readings and assignments is available on this page.
 * Participants are expected to start a web blog for the course and provide the address for their course blog on the participants page.
 * The course assignments are submitted on the participant's blog.
 * Later in a course the participants are asked to work on some content of their assignments on LeMill or Wikiversity and report this in their blog.
 * Participants should read the assigned material and post answers or responses to their blogs based on each week's prompting question.
 * Participants should use their blog to demonstrate understanding and critical assessment of assigned readings, and include original thoughts and synthesis of other's ideas.
 * The assignment posts should be available in the participants blog every Friday.

Evaluation and Feedback
At the end of the course the facilitators (Teemu and Hans) will provide PASS, FAIL or COMPLEMENTARY WORK REQUIRED statements for each participant. The participants are expected to evaluate and give constructive feedback to each other through comments on other blogs and synthesis of other's ideas on their own blog.

Schedule

Week 1: Introduction - Short history of free and open educational resources -

 * Open Educational Resources: What they are and why do they matter? by Ilkka Tuomi
 * Read from the beginning the first 10 pages, chapters "Introduction", "Future learning environments" and "The origins of the OER movement". You may skip the last sub-chapter "Openness in computer-based learning architectures".
 * Continue reading from page 25, the chapters "Open resources" and "Defining OER".

Example open education projects:
 * Open University (UK) Open Content Initiative
 * Rice Connexions - (See also Richard Baraniuk talks about the Connexions project at Ted Talks)
 * MIT OpenCourseWare
 * LeMill.net
 * OpenCourseWare Consortium a collaboration of more than 200 higher education institutions and associated organizations
 * Open.Michigan The University of Michigan's collection of Open Educational Resources.
 * Wikiversity - you are there now (!)

Assignments:
 * You should write three (3) blog posts
 * Create a short, two paragraphs, blog post introducing yourself. You may tell in it who and where you are, and why you want to take this course. You may also use audio or video to present yourself. If you want to stay anonymous it is fine, but please share us why you want to take the course.
 * Study the article by Ilkka Tuomi and write a commentary to his text in your own blog.
 * Study the open education projects online and write a commentary about them in your blog.
 * Your blog posts should demonstrate your understanding of the main ideas in the article.
 * Don't just summarize the article but try to present in your post some insights and original thoughts.

Week 2: Introduction to LeMill and Wikiversity - LeMill or Wikiversity in your own study project of creating open educational resource -

 * Introduction to LeMill
 * Introduction to Wikiversity
 * How LeMill can be used in this course?
 * How Wikiversity can be used in this course?

Assignments:
 * You should register with LeMill or Wikiversity, contribute something in the form of constructive feedback in discussion forums, editing, or new page creation, then write one (1) blog post about the experience
 * After registering with LeMill or Wikiversity, create a user profile page with information about yourself.
 * Try to locate a group or project where people are working on some learning resource or study project that you could assist with, and join the group or project by notifying the other editors of your interest.
 * As well as joining an existing project, start your own new content or wiki-page resource in LeMill or Wikiversity.
 * Document your steps, discoveries and ideas in your blog. How did you complete the assignments and how do you feel about LeMill and/or Wikiversity before and after your initial investigations.
 * Your blog post should show familiarity with LeMill and/or Wikiversity.

Links and resources
 * First Monday, Volume 14, Number 2 - 2 February 2009: Leinonen, T - Learning in and with an open wiki project: Wikiversity's potential in global capacity building

Week 3: Philosophical Background - Some threads of thoughts behind the open educational resource movement -

From Wikipedia:
 * Enlightenment - The Age of Enlightenment
 * Science and "standing on the shoulders of giants"
 * Library Movement: Origins of the public library as a social institution
 * Free Adult Education: Popular Education and Folk High School
 * Free Software Movement

Assignments:
 * You should write one (1) blog post
 * Study the articles about enlightenment, practice of science, library movement, free adult education, folk high school and free software movement.
 * Your blog post should discuss about all these themes.
 * Try to show similarities and differences in the ideas behind the themes.

Week 4: Copyright and Alternatives - From free software to free content resources -

From Wikipedia: From the Web:
 * The Idea of Copyright - History of copyright law
 * Free Culture - Watch the presentation by Lawrence Lessig.
 * Licensing and Creative Commons - Towards a Global Learning Commons: ccLearn By Ahrash Bissell and Jamie Boyle.

Assignments:
 * You should write one (1) blog post
 * Study the article about history of copyright, watch the presentation and check out the article about global learning commons.
 * Your blog post should discuss about all these themes.
 * Try to elaborate why the copyright is as it is and why Lessig is claiming that we need "Free Culture". Try to also contextualize these ideas in your own work and country.

Week 5: Wikipedia and Wikimedia - Free haven of free, libre and open content online -

 * How and Why Wikipedia Works: An Interview with Angela Beesley, Elisabeth Bauer, and Kizu Naoko by Dirk Riehle.
 * The Hidden Order of Wikipedia by Fernanda B. Viégas, Martin Wattenberg and Matthew M. McKeon.
 * Wikimedia projects

Assignments:
 * You should write one (1) blog post
 * Study the articles about Wikipedia and get familiar with the Wikimedia projects.
 * Your blog post should discuss about all these themes.
 * Try to elaborate what kind of impact the free culture and Wikimedia movements will have on educational resources and future of education in general.

Week 6: Taking and Sharing Pictures

- A picture says a thousand words -

How to shoot better photos?
 * Digital Photography Composition Tips - Composition tips by Digital Photography School
 * Quality images guidelines - Image quality guidelines in Wikimedia Commons

How to edit your digital photos?
 * Adobe Photoshop Express - web-based image editing software
 * Gimp - powerful free image editor for Windows, Mac and Linux
 * Paint.NET - image editor for Windows
 * Picasa - photo management and editing software by Google
 * Seashore - free image editor for Mac

Web sites for sharing photos under open licenses:
 * Wikimedia Commons - open photo and media database by Wikimedia Foundation
 * Flickr - popular photo sharing site owned by Yahoo, only part of the images are under Creative Commons licenses
 * CC Wiki - Featured Image Sites - image sites using Creative Commons licenses

Assignments:
 * You should write only one (1) blog post but do many other things, too
 * Study the materials / links.
 * Take a digital camera and shoot some photos in your home town. Choose one or two pictures that can be used in learning resources.
 * Do some basic editing for the photos that you have chosen (crop, correct exposure and white balance, add saturation, sharpen). You can use Photoshop Express or some other image editor (some suggestions above).
 * Join Flickr and upload the photos that you selected. Describe photos with tags, place them on a map and add a Creative Commons license.
 * Try to upload one photo to Wikimedia Commons or LeMill.
 * Search for your home town photos from Wikimedia Commons and Flickr. Did you find similar photos than yours? Were these under some open license?
 * Write a blog post where you summarize what you learned from the task. Add links to the photos that you have uploaded and found online.

Week 7: Sharing Audio

- And music can fill your heart with Love -

 * How to share your audios as educational resources online?
 * With this you need to look at installing audacity and LAME. (You won't need to install these two if you already have an audio recorder on your computer that produces MP3 files)
 * there is a good YouTube tutorial for this http://www.youtube.com/watch?v=r5K1ZsoO1sU
 * Once you have the software installed the fun begins; you can find a number of audacity tutorials on YouTube
 * After creating a few MP3 podcasts you can upload them to some of the podcasting services like;
 * Odeo
 * Switchpod
 * We need to find some Open podcast hosting sites
 * read about podcasting
 * read about Vorbis Ogg, it is an important part of openness and sound files
 * read about the creative commons and their connection to audio. Consider using CC licensed music files in your OER.

Assignments:
 * You should write only one (1) blog post but do many other things, too
 * Study the materials
 * Install audacity and LAME (or your favorite audio recorder)
 * Create a podcast that provides a detailed (and creative) description of the view outside your home, take a photograph of this view.
 * Upload the photograph to flickr
 * Upload the audio to odeo or switchpod
 * Include the photo and podcast as links in this week's blog posting, so we can all appreciate the view from your home
 * Think about how RSS fits into podcasting; later in the day describe RSS to a friend over coffee (or a free beer ;)

Week 8: Sharing Videos

 * Websites to find video (Note - not all videos on these sites are usable in OER):
 * Vidipedia - the free video encyclopedia that anyone can edit
 * Video Lectures - On demand and free video from the world's leading scientists
 * TeacherTube - Videos for teachers by teachers
 * Archive.org - Olden style videos, with an increasing number of new ones
 * Blip.tv -
 * Sclipo - Broadcast your skills
 * Expert Village - How to videos
 * Youtube - A lot of video
 * Sutree - Collecting how to videos from many other sites
 * Graspr - Instructional videos
 * Scivee - Research published on video
 * Videojug - Explaining videos

 * Making video more accessible - what about people who don't have broadband, or who can't hear, or who need it in text?
 * Tutorials for making videos - Windows and Macintosh
 * KinoDV - Editor for GNU Linux
 * Websites to remix video - with the necessary bandwidth, web based editors can solve the problem of not having video editing software on your own computer.

Assignments:
 * Find a video that you find useful from one or more of the sites listed above and embed the video in your blog. Embedding video is a simple way of displaying a video from one of the sites directly in your blog. When you find a video, look for the "embed" code and copy paste it to your blog.
 * Create your own 3-5 minute video and post it to Blip.tv and Youtube and then embed it on your blog
 * Post that same video to Dotsub and use that service to create subtitles for your video. Embed the new subtitled video on your blog
 * Many videos on the sites listed above do not offer copyrights for the videos they offer, or they do not give users the option to use open licensing like Creative Commons. Some of these sites do however! Write a post on your blog that points out the video sites that do contain openly licensed videos and suggest a process that you might go through to negotiate the open use of videos that you find that do not already have open license rights.

Week 9: Evaluation and Feedback

- Tell the others what do you think about their work -

Assignments:
 * You should write only one (1) blog post but do many other things, too
 * Visit each other's blogs and comment the posts in there.
 * Comment at least one blog post in each participant's blogs.
 * Write your own reflective thoughts about the course and the class. What did you learn?

Week 10: Presentations and Feedback

- How we can make it better next time -

Assignments:
 * You should write only one (1) blog post but write more to the Critique and feedback -page
 * Write also your own critique and feedback to the final blog post of the course / class.
 * Take part in the video-conference announced later in the facilitator's course/class blog

Acknowledgments

 * The course is sponsored by the The School of Art Education, University of Art and Design Helsinki.

 * Introduction to Open Education course by David Wiley has influence to the design of this course. Thank you!


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license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: File:2010 09 14 15 00 52.djvu


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User:Egm6321.f10.team1.allen/HW1

Given
We are given a Legendre differential equatons with n=1 as shown below.

$$L_2(y)=(1-x)^2\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+2y=0 $$

Both of the following homogeneous solutions satisfy the above differential equation.

$$ y_H^1=x=P_1(x) $$

$$ y_H^2=\frac{x}{2} \log \left( \frac{1+x}{1-x} \right ) - 1 = Q_1(x) $$

Find
We are asked to verify that

$${{L}_{2}}(y_{H}^{1})={{L}_{2}}(y_{H}^{2})=0$$.

Solution
In order to verify these solutions we must first obtain their first and second derivatives.

$$y_H^1(x)=P_n(x)=x$$

$$\frac{dy_H^1(x)}{dx}=1$$

$$\frac{d^2y_H^1(x)}{dx^2}=0$$

These solutions are plugged into the Legendre Differential Equation.

$$L_2(y_H^1)=(1-x)^2(0)-2x(1)+2(x)=0 $$

After simplifying.

$$L_2(y_H^1)=-2x+2x=0 $$

After cancelling terms, we obtain our desired results.

$${{L}_{2}}(y_{H}^{1})=0$$.

$$y_H^2(x)=Q_n(x)=\frac{x}{2} \log \left( \frac{1+x}{1-x} \right ) - 1$$

$${{L}_{2}}(x)={{L}_{2}}({\frac{x}{2} \log \left( \frac{1+x}{1-x} \right ) - 1})=0$$.

Author
Allen

title: User:Egm6321.f10.team1.allen/HW1


text:
User:PersonalGenotypingExperiment

I am about to undergo genotyping through 23andme, screening for some 500,000 SNPs. I intend to publish the entirety of the data online under Creative Commons license (possibly CC-0). My hope in doing so is that my genetic information may, in a tiny way, contribute to scientific understanding the human genome.

In aggregate, a large number of human genomes will help the general understanding of science. That dataset needs to be generated, so I'm happy to make my genome available.

In addition to releasing my raw data, there is supplemental information I can provide to potentially increase the usefulness of my genetic data. First and foremost, I will provide detailed information about my medical history and my individual traits. Additionally, since I have a background in biological research, I will document my own insights and conjectures about theoretical relationships between my traits and my genome.

One important observation is that any supplemental I provide should be released prior to undergoing genetic testing. At this moment, I am 'blind' to my genotypes. Once I learn receive the results of genetic test, those insights might affect my subsequent assessments and predictions, leading to potential confirmatory bias. Thus, to maximize the future usefulness of my genome results, I should document, insofar as possible, my state and trait insights prior to learning the results of genetic testing.

Wikiversity can, in this case, act as a trusted third party to document the drafting process and timeline. --PersonalGenotypingExperiment 02:13, 15 September 2010 (UTC)

The infinite complexity relationship genotype, environment, and disease
In approximately 8 weeks, I will receive a massive amount of data. Along with that data, 23andme will provide some very basic feedback in terms of what insight existing research to say about my genome. This information can be used in ways similar to a family history-- to potentially provide people with insights into what diseases they might develop in the future, allowing people to take appropriate steps to prevent those diseases.

That's a valid purpose, but it's not my purpose here. Rather, I hope my genetic results (combined with medical history and personal insights) can help us, as a species, gain a better understanding of the human being.

The personal genomic era is only just dawning now, and Epigenetics is an incredibly young science. Except in a handful of simple diseases, we are still a long, long way from true understandings, on a causal level, how genes and environment interact to cause disease.

So, upon learning my genotype, it will not be a simple matter to see the relationships between my gene results and my traits and diseases. Far, far from it. New discoveries about this relationship will continue to be made for my entire lifetime. We do not have a "perfect understanding" of this relationship, and Science will take decades or even centuries before arriving at such a complete understanding.

The need for a basic strategy
A human genotype constitutes a massive amount of data. The total genome is . Finding the genes that have contributed to my traits and my diseases is harder than finding needle in a haystack-- it's like trying to find specific set of haystalks hidden in a giant piles of bales of hay.

In order to help "narrow the search", I'll summarize my a priori knowledge and expectations. Specifically:
 * My notable traits, conditions, and diseases
 * Family history
 * Which traits or disease I suspect of having a genetic component
 * Which classes of systems are suspected of implication
 * Which classes of proteins might be implicated
 * Which families of genes might be implicated

In short, I'll try to help narrow the focus of where in my genotype a result should be considered "potentially significant".

Why guess?
In practice though, science is done in aggregate, not at an individual level. I sincerely may be of some small help to the progress of science by making my genome and my verifiable medical conditions available to the world.

Adding my own rough guesses of which systems and which genes might be implicated, however, probably won't be of any actual usefulness. In that way, they are the "human interest" side of the story. It's firstly and foremostly interesting to the human involved.

Every disease is a mystery, and the human brain loves mysteries. I look forward to watching science unravel the mysteries of our universe. Thus, it inspires me to participate-- to write down what my 'best guesses' about my genes and traits are. But even as I document such guesses, I know that this is only the opening pages of the mystery novel. With time, science will progress and reveal, piece by piece, how genes and traits are related. In the interim, I find it meaningful and enjoyable to speculate, even as I realize my guesses are only guesses, not evidence or answers. In a mystery novel, the reader starts trying to unwravel the mystery, beginning on page one. Fans of "Lost" began to speculate even after the first episode.

In a similar manner, I will engage in some guesses and speculation-- even though I know I do not yet have access to enough information to actually "solve the mystery". The era of personal genomics has only just begun to dawn.

Continued on User:PersonalGenotypingExperiment/Very Rough Draft

title: User:PersonalGenotypingExperiment


text:
User talk:Ffdssa/Mechation

Mechation
This site is particularly universal in principle, helping the world citizens to learn. Therefore, the simpler vocabulary the better. As a utilitarian who admired Jeremy Bentham in particular, C. K. Ogden deduced the Basic English (1930) comprising only 850 words so as to be easily used by them. This was welcome by H. G. Wells who wished them to be educated and integrated into the World Brain (1938), something like this wiki complex.

In this regard, neologisms are implausible in principle. Some of them may even do harm to the rhetoric. The term 'mechation' may give way to 'inorganic evolution' or the like to be more decent. Symbols live to evolve by natural selection, too, struggling for survival! The weak are ruthlessly weeded out. The worst is to reinvent neologisms synonymous to the existing, whether piratingly or unwittingly.

The Pirate Party has become explodingly popular in many western countries in four years. Such copy-leftism should be counter-balanced by, so to speak, 'idea-rightism' that aims to pay due nominal (rather than financial) homage to the original idea (or Platonic Idea) or conception. This is a must to establish the orthodox lineage of scientific paradigms, while weeding out too many charlatans disguised by private synonyms, contributing to maximazing entropy.

-- KYPark [T] 11:11, 15 September 2010 (UTC)

Signs and Symbols
Signs are whatever is interpreted, according to Ogden & Richards (1923) The Meaning of Meaning, or simply whatever is (e)valuated, of either nature or culture, in my view. So they include socio-cultural symbols. Animals are supposed to use or interpret signs and even symbols as per their primitive capacity. Humans have evolved just far away from them only to be too dependent on symbols to be free from lies and other harms. Evolution goes hand in hand with ever-increasing entropy or complexity.

Artefacts or creations also serve as symbols. Complicating the human community, either may be a curse as well as blessing, from the ultimate entropic perspective, say, toward death. Humans are destined to analytically refer their attention to the past experiences in terms of sign and symbol, and then synthetically infer their intensive intention or creative conception of values as well as more symbols from them. This is the way they have been evolving to live by in context. And they are getting more and more trapped and choked by their own creatures, perhaps including God!

Humans are either animals in a way or not in another, as suggested by the name of God. So many (most at worst) of them are schizo, suffering double bind. This single perspective may be enough to be extremely pessimistic about arriving at any reasonable consensus on human destiny. Furthermore discouragingly, the opening passage of the above-mentioned book (1923) includes the following quotation from Lao Tse:
 * "He who knows does not speak, he who speaks does not know."

-- KYPark [T] 05:43, 15 September 2010 (UTC)

The Stages of Mechation
The above TOC is intended as a place for discussion from section to section. -- KYPark [T] 03:43, 15 September 2010 (UTC)

title: User talk:Ffdssa/Mechation


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User talk:PersonalGenotypingExperiment

 Hello PersonalGenotypingExperiment, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 04:01, 15 September 2010 (UTC)

title: User talk:PersonalGenotypingExperiment


text:
User:Egm6321.f10.team03/Hwk2

=Homework Two Summary=

= Problem 1 - Non-linearity of in general N1_ODE =

Given
Given is the in general form of a N1_ODE:
 * {| style="width:100%" border="0"

$$M(x,y)+N(x,y)\frac{dy}{dx}=0$$
 * $$\,(Eq. 1.1) $$
 * style= |
 * }

Find
It has to be shown, that IN GENERAL (Eq. 1.1) is non-linear

Solution
To show that (Eq. 1.1) is in general non-linear it needs to meet the non-linearity condition, which is:
 * {| style="width:100%" border="0"

$$F\left( \alpha u+\beta v \right)\ne F\left( \alpha u \right)+F\left( \beta v \right)=\alpha F\left( u \right)+\beta F\left( v \right)$$ Applying the condition above to (Eq. 1.1), we have to see if the following equation is true:
 * $$\,(Eq. 1.2) $$
 * style= |
 * }
 * {| style="width:100%" border="0"

$$cM(x,y)+cN(x,y)\frac{dy}{dx}\overset{?}{\mathop{=}}\,M(x,cy)+N(x,cy)\frac{d(cy)}{dx}$$, where $$c$$ is constant
 * $$\,(Eq. 1.3) $$
 * style= |
 * }

Now, since N(x,y) can be ANY function, like a cos(2xy) which is non-linear, it turns out that (Eq. 1.3) is not true and is IN GENERAL not linear since:
 * {| style="width:100%" border="0"

$$cM(x,y)+cN(x,y)\frac{dy}{dx}\ne M(x,cy)+N(x,cy)\frac{dcy}{dx}=M(x,cy)+cN(x,cy)\frac{dy}{dx}$$ So it can be said that: :{| style="width:30%" border="0" $$M(x,y)+N(x,y)\frac{dy}{dx}=0\ is\ in\ general\ non\ linear!$$ For particular solutions it might be the case, that (Eq. 1.1) is linear, but a particular solution does not describe the most general case of an equation.
 * $$\,(Eq. 1.4) $$
 * style= |
 * }
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * }
 * }

Contributing Members

 * Solved by Michele Sigilló
 * Posted by Michele Sigilló
 * Solved and Proof read by Egm6321.f10.team3.Sudheesh 17:01, 20 September 2010 (UTC)
 * Solved and Proof read by Adam Franklin 20:23, 20 September 2010 (UTC)

= Problem 2 - Verification of an equation to be a N1-ODE=

Given

 * {| style="width:100%" border="0"

$$ 4 {x}^{7} + \sin\left(y\right) +  {x}^{2} {y}^{3} {y}^{'} = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * $$\displaystyle (7) p. 7-1
 * }

Find
Verify that $$ 4 {x}^{7} + \sin\left(y\right) +  {x}^{2} {y}^{3} {y}^{'} $$ is a N1-ODE

Solution
For $$\displaystyle (7) p. 7-1$$ (shown above in "Given") to be linear it must obey the following conditions for linearity given below:
 * {| style="width:100%" border="0"

f(cx) = cf(x) $$ where $$ c \in \mathbb{R}$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq 2.1)
 * style= |
 * }

 * {| style="width:100%" border="0"

f(x+y) = f(x) + f(y) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.2.2)
 * style= |
 * }

To show that $$\displaystyle (7) p.7-1$$ is nonlinear we will use $$\displaystyle (Eq 2.1)$$ and show that this condition is not true

 * {| style="width:100%" border="0"

$$
 * Let $$ \mathrm{L} \left( y\right):= 4 {x}^{7} + sin \left( y\right) + {x}^{2} {y}^{3} {y}^{'}$$
 * $$\displaystyle (Eq.2.3)
 * $$\displaystyle (Eq.2.3)
 * style= |
 * }

 * {| style="width:100%" border="0"

$$
 * $$ \mathrm{L} \left( cy\right)= 4 {x}^{7} + sin \left( cy\right) + {x}^{2} {\left(cy\right)}^{3} \frac{\ {d} }{\ {d} x} \left( cy\right)$$
 * $$\displaystyle (Eq.2.4)
 * $$\displaystyle (Eq.2.4)
 * style= |
 * }

 * {| style="width:100%" border="0"

$$ But
 * $$ \mathrm{L} \left( cy\right)= 4 {x}^{7} + sin \left( cy\right) + {c}^{4}{x}^{2} {y}^{3} \frac{\ {d} }{\ {d} x} \left( y\right)$$
 * $$\displaystyle (Eq.2.5)
 * $$\displaystyle (Eq.2.5)
 * style= |
 * }
 * {| style="width:100%" border="0"

$$
 * $$ c\mathrm{L} \left( y\right)= c \left( 4 {x}^{7} + sin \left( y\right) + {x}^{2} {y}^{3} {y}^{'}\right)$$
 * $$\displaystyle (Eq.2.6)
 * $$\displaystyle (Eq.2.6)
 * style= |
 * }

 * {| style="width:100%" border="0"

$$ \Rightarrow 4 {x}^{7} + sin \left( cy\right) + {c}^{4}{x}^{2} {y}^{3} {y}^{'}\neq c \left( 4 {x}^{7} + sin \left( y\right) + {x}^{2} {y}^{3} {y}^{'}\right)$$ By inspection, $$ \mathrm{L} \left( cy\right)\neq c\mathrm{L} \left( y\right)$$
 * style="width:25%; padding:1px; border:2px solid #0000FF" |
 * style="width:25%; padding:1px; border:2px solid #0000FF" |
 * style= |
 * }
 * }

Also by inspection, $$\displaystyle (7) p. 7-1$$ is first order because the equation contains the term $$\displaystyle y'$$ but not $$\displaystyle y''$$, which would make the equation 2nd order.

This verifies that $$ 4 {x}^{7} + \sin\left(y\right) +  {x}^{2} {y}^{3} {y}^{'} $$ is an N1-ODE

Contributing Members

 * Solved and posted by --Egm6321.f10.team3.cook 23:35, 19 September 2010 (UTC)
 * Proofed by Kurt Schulze
 * Proof read by : Egm6321.f10.team03.sigillo 01:04, 21 September 2010 (UTC)
 * Proof read by : Adam Franklin 20:04, 21 September 2010 (UTC)

=Problem 3: Linear independence of two homogeneous solutions=

Given
Given a linear second order differential equation with varying coefficient of the form:

 * {|style="width:100%" border="0"

$$\,{a}_{2} \left( x\right)\ddot{y}+{a}_{1} \left(x \right) \dot{y}+{a}_{0} \left( x\right)y=f\left(x \right)$$
 * $$\,(Eq.3.1)$$
 * }

Assume $$L_2=f \left( x \right)$$, a superposition of solution can be written as:

 * {|style="width:100%" border="0"

$$\,L_2\left( y_H \right)+L_2 \left( y_P\right)=f \left( x\right)$$
 *
$$\,(Eq.3.2)$$
 * }

where
 * {|style="width:100%" border="0"

$$\,y_{H}=c \left( x\right)+d \left( x \right)$$
 *
$$\,(Eq.3.3)$$
 * }

where
 * {|style="width:100%" border="0"

$$\,{y^1_{H}} \left( x\right)=x$$ and
 *
$$\,(Eq.3.4)$$
 * }
 * {|style="width:100%" border="0"

$$\,{y^2_{H}} \left( x\right)= \frac{x}{2} \log \left(  \frac{1+x}{1-x}\right)-1$$
 *
$$\,(Eq.3.5)$$
 * }

Find
Plot functions $${y^1_{H}} \left( x\right)$$ and $${y^2_{H}} \left( x\right)$$ and show linear independence between the two functions.

Solution
In general, linear dependence among n number of vectors can be defined as $$a_1 \mathbf{v}_1 + \cdots + a_n \mathbf{v}_n=0 $$. In this case, linear independence can be defined as

 * {|style="width:100%" border="0"

$$\,{y^1_{H}} \left( x\right) \neq \alpha{y^2_{H}} \left( x\right)$$ or there exists no constant $$\alpha$$ that makes equations 3.4 and 3.5 equivalent for all $$\alpha$$ and for all $$x$$ in the real domain. To prove this, let us assume:
 *
$$\,(Eq.3.6)$$
 * }

 * {|style="width:100%" border="0"

$$\,\alpha= \frac{{y^1_{H}}\left( x\right)}{{y^2_{H}} \left( x\right)}$$
 *
$$\,(Eq.3.7)$$
 * }

Evaluating equation 3.7 at $$x=\frac{1}{2}$$ gives us $$\alpha=-0.6893$$.

 * {|style="width:100%" border="0"

$$\,{y^1_{H}} \left( x\right) =\alpha{y^2_{H}} \left( x\right)$$ Evaluating equation 3.8 with the assumed value of $$\alpha$$ at $$x=\frac{1}{4}$$ yields:
 *
$$\,(Eq.3.8)$$
 * }

 * {|style="width:100%" border="0"

$$.25 \neq 0.6453$$
 *
$$\,(Eq.3.9)$$
 * }
 * FunctionsY1H_and_Y2H.jpg

The values are not equivalent for all $$x$$ in the domain, therefore the two functions are linearly independent. Further more, figure 3.1 agrees with this conclusion.

Contributing Members

 * Solved and authored by James Roark 12:32, 18 September 2010 (UTC)
 * Proof read by: Egm6321.f10.team03.sigillo 01:00, 21 September 2010 (UTC)

= Problem 4 - Create Exact Nonlinear First Order Ordinary Differential Equation (N1-ODE) =

Given

Conditions of Exactness:

 * For function $$F\left( . \right)$$ to be exact:

 * $$\left( 1 \right)$$ $$F\left( . \right)$$ must be in the form:

 * {| style="width:100%" border="0"

$$
 * $$F\left( x,y,{{y}^{1}} \right)=M\left( x,y \right)+N\left( x,y \right){{y}^{'}}$$
 *
$$\displaystyle (Eq. 4.1)
 *
$$\displaystyle (Eq. 4.1)
 * }
 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

 * Where $$M\left( x,y \right)$$ and $$N\left( x,y \right)$$ are functions of $$\,x$$  and $$\,y$$ and where $$=\frac{dy}{dx}$$
 * }
 * }
 * }

 * $$\left( 2 \right)$$ $$M\left( x,y \right)$$, and $$N\left( x,y \right)$$ must satisfy:

 * {| style="width:100%" border="0"

$$
 * $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$
 *
$$\displaystyle (Eq. 4.2)
 *
$$\displaystyle (Eq. 4.2)
 * }
 * }

Given function:

 * {| style="width:100%" border="0"

$$
 * $$\phi \left( x,y \right)={{x}^{2}}{{y}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}}+\log \left( {{x}^{3}}{{y}^{2}} \right)=k$$
 *
$$\displaystyle (Eq. 4.3)
 *
$$\displaystyle (Eq. 4.3)
 * }
 * }

Find

 * $$\left( 1 \right)$$ Find function $$F\left( . \right)$$ such that $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}$$

 * $$\left( 2 \right)$$ Verify that $$F\left( x,y,{{y}^{1}} \right)$$ is an Exact N1-ODE

 * $$\left( 3 \right)$$ Invent three more Exact N1-ODE

Solution

 * $$\left( 1 \right)$$ Find function $$F\left( x,y,{{y}^{1}} \right)$$:

 * Derivative of $$\phi \left( x,y \right)$$ is given by
 * {| style="width:100%" border="0"

$$
 * $$\frac{d\phi }{dx}=\frac{\partial \phi }{\partial x}+\frac{\partial \phi }{\partial y}\cdot {{y}^{'}}$$
 *
$$\displaystyle (Eq. 4.4)
 *
$$\displaystyle (Eq. 4.4)
 * }
 * }

 * {| style="width:100%" border="0"

$$
 * $$\frac{\partial \phi }{\partial x}=\left[ 2x{{y}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}}+\frac{1}\cdot 3{{x}^{2}}{{y}^{2}} \right]=2x{{y}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}}+\frac{3}{x}\Rightarrow M(x,y)$$
 *
$$\displaystyle (Eq. 4.5)
 *
$$\displaystyle (Eq. 4.5)
 * }
 * }

 * {| style="width:100%" border="0"

$$
 * $$\frac{\partial \phi }{\partial y}=\left[ \frac{3}{2}{{x}^{2}}{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{1}\cdot 2{{x}^{3}}y \right]=\frac{3}{2}{{x}^{2}}{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{2}{y}\Rightarrow N(x,y)$$
 *
$$\displaystyle (Eq. 4.6)
 *
$$\displaystyle (Eq. 4.6)
 * }
 * Substituting Eq.4.5 & 4.6 into Eq.4.4 yields:
 * Substituting Eq.4.5 & 4.6 into Eq.4.4 yields:

 * {| style="width:100%" border="0"

$$
 * $$\frac{d\phi }{dx}=\left[ 2x{{y}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}}+\frac{3}{x} \right]+\left[ \frac{3}{2}{{x}^{2}}{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{2}{y} \right]\cdot {{y}^{'}}$$
 *
$$\displaystyle (Eq. 4.7)
 *
$$\displaystyle (Eq. 4.7)
 * }
 * Therefore:
 * Therefore:

 * {| style="width:100%" border="0"

$$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}=\underbrace{\left[ 2x{{y}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}}+\frac{3}{x} \right]}_{M\left( x,y \right)}+\underbrace{\left[ \frac{3}{2}{{x}^{2}}{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{2}{y} \right]}_{N\left( x,y \right)}\cdot {{y}^{'}}$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 *
$$\displaystyle (Eq. 4.8)
 * }
 * }

 * $$\left( 2 \right)$$ Verify that $$F\left( x,y,{{y}^{1}} \right)$$ is an Exact N1-ODE:

 * The function $$F\left( x,y,{{y}^{1}} \right)$$ is a first order nonlinear differential equation as its highest order is 1 and does not satisfy the condtion of superposition (linearity).
 * The function $$F\left( x,y,{{y}^{1}} \right)$$ given in Eq. 4.8 has the structure of Eq.4.1 and hence first condition of Exactness is satisfied.
 * Definiton of the second condition of exactness yields to:
 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

$$
 * $$\frac{\partial M}{\partial y}=\frac{\partial }{\partial y}\left( 2x{{y}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}}+\frac{3}{x} \right)=2x\cdot \frac{3}{2}{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+0=3x{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}$$
 *
$$\displaystyle (Eq. 4.9)
 *
$$\displaystyle (Eq. 4.9)
 * }
 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

$$
 * $$\frac{\partial N}{\partial x}=\frac{\partial }{\partial x}\left( \frac{3}{2}{{x}^{2}}{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{2}{y} \right)=\frac{3}{2}\cdot 2x{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+0=3x{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}$$
 *
$$\displaystyle (Eq. 4.10)
 *
$$\displaystyle (Eq. 4.10)
 * }
 * Eq.4.9 and 4.10 implies,
 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

$$
 * $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$
 *
$$\displaystyle (Eq. 4.11)
 *
$$\displaystyle (Eq. 4.11)
 * }
 * Eq.4.11 satisfy the second condition of Exactness.
 * Eq.4.11 satisfy the second condition of Exactness.

 * Since both conditions of exactness are satisifed,
 * {| style="width:100%" border="0"

 * $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}=\underbrace{\left[ 2x{{y}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}}+\frac{3}{x} \right]}_{M\left( x,y \right)}+\underbrace{\left[ \frac{3}{2}{{x}^{2}}{{y}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{2}{y} \right]}_{N\left( x,y \right)}\cdot {{y}^{'}}$$ is an Exact N1-ODE
 * }
 * $$\left( 3 \right)$$ Invent three more Exact N1-ODE:
 * Example 1:
 * Consider, $$\phi \left( x,y \right)=3x\cdot {{e}^{y}}+4{{y}^{2}}\cdot {{e}^{x}}= k $$
 * Example 1:
 * Consider, $$\phi \left( x,y \right)=3x\cdot {{e}^{y}}+4{{y}^{2}}\cdot {{e}^{x}}= k $$

 * {| style="width:100%" border="0"

 * $$\frac{\partial \phi }{\partial x}=3\cdot {{e}^{y}}+4{{y}^{2}}\cdot {{e}^{x}}\Rightarrow M(x,y)$$
 * }
 * {| style="width:100%" border="0"
 * }
 * {| style="width:100%" border="0"

 * $$\frac{\partial \phi }{\partial y}=3x\cdot {{e}^{y}}+8y\cdot {{e}^{x}}\Rightarrow N\left( x,y \right)$$
 * }
 * Now consider a function $$F\left( x,y,{{y}^{1}} \right)$$ such that $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}$$
 * }
 * Now consider a function $$F\left( x,y,{{y}^{1}} \right)$$ such that $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}$$

 * {| style="width:100%" border="0"

 * $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}=M\left( x,y \right)+N\left( x,y \right)\cdot {{y}^{'}}=0$$
 * }
 * {| style="width:100%" border="0"
 * }
 * {| style="width:100%" border="0"

$$F\left( x,y,{{y}^{1}} \right)=\left[ 3\cdot {{e}^{y}}+4{{y}^{2}}\cdot {{e}^{x}} \right]+\left[ 3x\cdot {{e}^{y}}+8y\cdot {{e}^{x}} \right]\cdot {{y}^{'}}=0$$ $$
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 *
$$\displaystyle (Eq. 4.12)
 * }
 * This is an Exact N1-ODE
 * This is an Exact N1-ODE

 * Example 2:
 * Consider, $$\phi \left( x,y \right)=5y\log \left( {{x}^{2}} \right)+3{{x}^{2}}\sin \left( {{y}^{3}} \right)=k$$

 * {| style="width:100%" border="0"

 * $${{\phi }_{x}}\left( x,y \right)=5y\cdot \left( \frac{1}\cdot 2x \right)+6x\sin \left( {{y}^{3}} \right)=10\frac{y}{x}+6x\sin \left( {{y}^{3}} \right)\Rightarrow M(x,y)$$
 * }
 * {| style="width:100%" border="0"
 * }
 * {| style="width:100%" border="0"

 * $${{\phi }_{y}}\left( x,y \right)=5\log \left( {{x}^{2}} \right)+3{{x}^{2}}\left( \cos \left( {{y}^{3}} \right)\cdot 3{{y}^{2}} \right)=5\log \left( {{x}^{2}} \right)+9{{x}^{2}}{{y}^{2}}\cos \left( {{y}^{3}} \right)\Rightarrow N\left( x,y \right)$$
 * }
 * Now consider a function $$F\left( x,y,{{y}^{1}} \right)$$ such that $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}$$
 * }
 * Now consider a function $$F\left( x,y,{{y}^{1}} \right)$$ such that $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}$$

 * {| style="width:100%" border="0"

 * $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}=M\left( x,y \right)+N\left( x,y \right)\cdot {{y}^{'}}=0$$
 * }
 * {| style="width:100%" border="0"
 * }
 * {| style="width:100%" border="0"

$$F\left( x,y,{{y}^{'}} \right)=\left[ 10\frac{y}{x}+6x\sin \left( {{y}^{3}} \right) \right]+\left[ 5\log \left( {{x}^{2}} \right)+9{{x}^{2}}{{y}^{2}}\cos \left( {{y}^{3}} \right) \right]{{y}^{'}}=0$$ $$
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 *
$$\displaystyle (Eq. 4.13)
 * }
 * This is an Exact N1-ODE
 * This is an Exact N1-ODE

 * Example 3:
 * Consider, $$\phi \left( x,y \right)=7{{x}^{3}}\cdot \cos \left( {{y}^{3}} \right)+2{{y}^{2}}{{e}^{{{x}^{2}}\cdot \sin y}}=k$$

 * {| style="width:100%" border="0"

& {{\phi }_{x}}\left( x,y \right)=7\times 3{{x}^{2}}\cdot \cos \left( {{y}^{3}} \right)+2{{y}^{2}}\cdot {{e}^{{{x}^{2}}\cdot \sin y}}\cdot \left( 2x\cdot \sin y \right)=21{{x}^{2}}\cdot \cos \left( {{y}^{3}} \right)+4x{{y}^{2}}\cdot \sin y\cdot {{e}^{{{x}^{2}}\cdot \sin y}}\Rightarrow M(x,y) \\ & \\ \end{align}$$
 * $$\begin{align}
 * $$\begin{align}

 * }
 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

 * $${{\phi }_{y}}\left( x,y \right)=7{{x}^{3}}\cdot \left[ -\sin \left( {{y}^{3}} \right)\cdot 3{{y}^{2}} \right]+2\left[ 2y\cdot {{e}^{{{x}^{2}}\cdot \sin y}}+{{y}^{2}}\cdot {{e}^{{{x}^{2}}\cdot \sin y}}\cdot {{x}^{2}}\cdot \cos y \right]=-21{{x}^{3}}{{y}^{2}}\sin \left( {{y}^{3}} \right)+4y{{e}^{{{x}^{2}}\cdot \sin y}}+2{{x}^{2}}{{y}^{2}}\cos y\cdot {{e}^{{{x}^{2}}\cdot \sin y}}\Rightarrow N\left( x,y \right)$$
 * }
 * Now consider a function $$F\left( x,y,{{y}^{1}} \right)$$ such that $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}$$
 * }
 * Now consider a function $$F\left( x,y,{{y}^{1}} \right)$$ such that $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}$$

 * {| style="width:100%" border="0"

 * $$F\left( x,y,{{y}^{1}} \right)=\frac{d\phi }{dx}=M\left( x,y \right)+N\left( x,y \right)\cdot {{y}^{'}}=0$$
 * }
 * {| style="width:100%" border="0"
 * }
 * {| style="width:100%" border="0"

$$\left( x,y,{{y}^{'}} \right)=\left[ 21{{x}^{2}}\cdot \cos \left( {{y}^{3}} \right)+4x{{y}^{2}}\cdot \sin y\cdot {{e}^{{{x}^{2}}\cdot \sin y}} \right]+\left[ -21{{x}^{3}}{{y}^{2}}\sin \left( {{y}^{3}} \right)+4y{{e}^{{{x}^{2}}\cdot \sin y}}+2{{x}^{2}}{{y}^{2}}\cos y\cdot {{e}^{{{x}^{2}}\cdot \sin y}} \right]{{y}^{'}}=0$$ $$
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 *
$$\displaystyle (Eq. 4.14)
 * }
 * This is an Exact N1-ODE
 * This is an Exact N1-ODE

Contributing Members

 * Solved and posted by --Egm6321.f10.team3.Sudheesh 15:20, 20 September 2010 (UTC)
 * Proofed by Kurt Schulze
 * Proof read by: Egm6321.f10.team03.sigillo 01:39, 21 September 2010 (UTC)

= Problem 5 - Non exact N1_ODE=

Given
The general form of a N1_ODE is given as being:
 * {| style="width:100%" border="0"

$$F(x,y,{y}')=M(x,y)+N(x,y)f\left( {{y}'} \right)=0$$
 *
$$\,(Eq. 5.1) $$
 * style= |
 * }

Find
To be found is a function $$f\left( {{y}'} \right)$$ such that there is no analytical solution to $$f\left( {{y}'} \right)=\frac{-M}{N}$$ , hence a non exact N1_ODE has to be found.

Solution
Let's have a look at the chosen non-linear equation :
 * {| style="width:100%" border="0"

$$2x{{y}^{2}}+\cos \left( 3y \right)x\cdot {y}'\exp \left( {{y}'} \right)=0$$ where the three terms of the general equation (Eq. 5.1) in this application case can be found as:
 *
$$\,(Eq. 5.2) $$
 * style= |
 * }
 * {| style="width:100%" border="0"

$$M\left( x,y \right)=2x{{y}^{2}}\quad \quad N\left( x,y \right)=\cos \left( 3y \right)x\quad \quad f({y}')={y}'\cdot \exp \left( {{y}'} \right)$$ Since the first conditoin for an exact N1_ODE is defined to be:
 * style= |
 * }
 * }
 * {| style="width:100%" border="0"

$$\,M(x,y)+N(x,y){y}'=0$$ It's needed to put (Eq. 5.2) in the form of (Eq. 5.3). Thus the next step is to solve (Eq. 5.2) for y' :
 *
$$\,(Eq. 5.3) $$
 * style= |
 * }
 * {| style="width:100%" border="0"

$${y}'\cdot \exp \left( {{y}'} \right)=\frac{-2x{{y}^{2}}}{\cos \left( 3y \right)x}=\frac{-2{{y}^{2}}}{\cos \left( 3y \right)}$$ trying to get y' out of the exp-function we apply the logarithm-funtion:
 *
$$\,(Eq. 5.4) $$
 * style= |
 * }
 * {| style="width:100%" border="0"

$$\ln \left( {y}'\cdot \exp \left[ {{y}'} \right] \right)=\ln \left( {{y}'} \right)+\ln \left( \exp \left[ {{y}'} \right] \right)=\ln \left( \frac{-2{{y}^{2}}}{\cos \left( 3y \right)} \right)$$
 *
$$\,(Eq. 5.5) $$
 * style= |
 * }
 * {| style="width:100%" border="0"

$$\ln \left( {{y}'} \right)+{y}'=\ln \left( \frac{-2{{y}^{2}}}{\cos \left( 3y \right)} \right)$$ :{| style="width:90%" border="0" As it can be seen, there is on analytical solution for y' in (Eq. 5.6), this means that (Eq. 5.2) can not be put in the form of (Eq. 5.3), hence it is NOT an exact N1_ODE !
 *
$$\,(Eq. 5.6) $$
 * style= |
 * }
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * }
 * }

Contributing Members

 * Solved by Michele Sigilló
 * Posted by Michele Sigilló
 * Proof read by:Egm6321.f10.team3.Sudheesh 15:29, 20 September 2010 (UTC)

= Problem 6 - Verification of a particular solution y(x)=

Given
Suppose a function $$\phi $$ can be found to be a reduced order solution and its derivative is equal to:
 * {| style="width:100%" border="0"

$$\frac{d\phi (x,y,{y}')}{dx}=F(x,y,{y}')=75{{x}^{4}}+\cos \left( y \right){y}'=0$$ $$ and the explicit solution for $$\,y(x)$$ is given as:
 *
$$\,(Eq. 6.1)
 * style= |
 * }
 * {| style="width:100%" border="0"

$$y(x)={{\sin }^{-1}}\left( k-15{{x}^{5}} \right)$$. $$
 *
$$\,(Eq. 6.2)
 * style= |
 * }

Find
It has to be proven that (Eq. 6.2) satisfies (Eq. 6.1)

Solution
In general $$F(x,y,{y}')$$ can be written as:
 * {| style="width:100%" border="0"

$$\,F(x,y,{y}')=M(x,y)+N(x,y){y}'=0$$ $$ where from (Eq. 6.1) it is known that
 *
$$\,(Eq. 6.3)
 * style= |
 * }
 * {| style="width:100%" border="0"

$$\,M(x,y)=75{{x}^{4}}$$ $$ and
 *
$$\,(Eq. 6.4)
 * style= |
 * }
 * {| style="width:100%" border="0"

$$\,N(x,y)=\cos \left( y \right)$$ $$ The chosen approach is to derivate $$y(x)$$ ,insert the result into (Eq. 6.2) and check if it satisfies the condition $$F(x,y,{y}')=0$$.In order to have a better overview during deferentiating, let's define
 *
$$\,(Eq. 6.5)
 * style= |
 * }
 * {| style="width:100%" border="0"

$$u(x)=\left( k-15{{x}^{5}} \right)$$ $$ The derivative is defined to be:
 *
$$\,(Eq. 6.6)
 * style= |
 * }
 * {| style="width:100%" border="0"

$${y}'=\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$ $$ which in the particular case it is:
 *
$$\,(Eq. 6.7)
 * style= |
 * }
 * {| style="width:100%" border="0"

$$\frac{dy}{du}=\frac{1}{\sqrt{1-{{u}^{2}}}}$$ $$ and
 *
$$\,(Eq. 6.8)
 * style= |
 * }
 * {| style="width:100%" border="0"

$$\frac{du}{dx}=-75{{x}^{4}}$$ $$ Inserting (Eq. 6.8) and (Eq. 6.9) into (Eq. 6.7) while substituiting $$u$$ back, (Eq. 6.7) can be rewritten as:
 *
$$\,(Eq. 6.9)
 * style= |
 * }
 * {| style="width:100%" border="0"

$${y}'=\frac{-75{{x}^{4}}}{\sqrt{1-{{\left( k-15{{x}^{5}} \right)}^{2}}}}$$ $$ The key-point of this Problem in terms of quickness, lays in the fact that is has to be recognized that $${{\left( k-15{{x}^{5}} \right)}^{2}}$$ is the same like $${{sin }^{2}}\left( y \right)$$, hence (Eq. 6.10) becomes:
 *
$$\,(Eq. 6.10)
 * style= |
 * }
 * {| style="width:100%" border="0"

$${y}'=\frac{-75{{x}^{4}}}{\sqrt{1-{{\sin }^{2}}\left( y \right)}}=\frac{-75{{x}^{4}}}{\sqrt{{{\cos }^{2}}\left( y \right)}}=\frac{-75{{x}^{4}}}{\cos \left( y \right)}$$ $$ Now substituting y' in (Eq. 6.1) with (Eq. 6.11) we get:
 *
$$\,(Eq. 6.11)
 * style= |
 * }
 * {| style="width:100%" border="0"

$$75{{x}^{4}}+\cos \left( y \right)\frac{-75{{x}^{4}}}{\cos \left( y \right)}=75{{x}^{4}}-75{{x}^{4}}=0$$ $$ :{| style="width:70%" border="0" Left-Hand-Side and Right-Hand-Side of the equation are the same, hence (Eq. 6.2) satisfies (Eq. 6.1) !
 *
$$\,(Eq. 6.12)
 * style= |
 * }
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * }
 * }

Contributing Members

 * Solved by Kurt Schulze
 * Posted by Michele Sigilló
 * Proof read by: Egm6321.f10.team03.sigillo 00:39, 21 September 2010 (UTC)

= Problem 7 - Solving a L1-ODE using Euler integrationfactor method=

Part 1
Solution to linear, first order, ordinary differential equation with varying coefficients using Euler integrating factor method

Given
Given

 * {|style="width:100%" border="0"

$$a_{1}\left( x \right){y}'+a_{0}\left( x \right)y=b\left( x \right)$$

 *
$$\,(Eq.7.1)$$
 * }
 * where
 * {|style="width:100%" border="0"

$$a_{1}\left( x \right)=1$$
 *
$$\,(Eq.7.2)$$
 * }

 * {|style="width:100%" border="0"

$$a_{0}\left( x \right)=x$$
 *
$$\,(Eq.7.3)$$
 * }

 * {|style="width:100%" border="0"

$$b\left( x \right)=2x+3$$
 *
$$\,(Eq.7.4)$$
 * }

Find
Find $$y$$ using Euler Integration Factor

Solution
Substituting equations 7.2, 7.3, and 7.4 into 7.1:

 * {|style="width:100%" border="0"

$$\,{y}'+xy=2x+3$$ Equation 7.5 is of the form $${y}'+P\left( x \right)y=Q\left( x \right)$$ The integrating factor for this form is $$I\left( x \right)=\exp \left[{P}\left( x \right)dx \right]$$. For equation 7.5, the integrating factor is
 *
$$\,(Eq.7.5)$$
 * }

 * {|style="width:100%" border="0"

$$I\left( x \right)=\exp \left[ \int{x}dx \right]=\exp \left[ \frac{x^{2}}{2} \right]$$
 *
$$\,(Eq.7.6)$$
 * }

Applying the integrating factor for equation 7.5:

 * {|style="width:100%" border="0"

$$\exp \left[ \frac{2} \right]\frac{dy}{dx}+\exp \left[ \frac{2} \right]xy=2x\exp \left[ \frac{2} \right]+3\exp \left[ \frac{2} \right]$$
 *
$$\,(Eq.7.7)$$
 * }

(Eq. 7.7) reduces to
 * {|style="width:100%" border="0"

$$\frac{d}{dx}\left( y\exp \left( \frac{2} \right) \right)=2x\exp \left[ \frac{2} \right]+3\exp \left[ \frac{2} \right]$$
 *
$$\,(Eq.7.8)$$
 * }

Integrating

 * {|style="width:100%" border="0"

$$y\exp \left( \frac{2} \right)=\int{2x\exp \left[ \frac{2} \right]}\,dx+\int{3\exp \left[ \frac{2} \right]}\,dx$$
 *
$$\,(Eq.7.9)$$
 * }

results in the following equation:
 * {|style="width:100%" border="0"

$$y\exp \left( \frac{2} \right)=2\exp \left[ \frac{2} \right]\,+\int{3\exp \left[ \frac{2} \right]}\,dx$$
 *
$$\,(Eq.7.10)$$
 * }

There is no known anti-derivative of the last term, so using mathematical analysis tools the last therm results in the error-function. So that rearranging the equation we get for y(x):

$$y=2+3\exp \left[ \frac{x^{2}}{2} \right]\left[ \sqrt{\frac{\pi }{2}} erfi\left( \frac{x}{\sqrt{2}} \right) \right]$$
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 *
$$\,(Eq.7.11)$$
 * }

Part 2
A general solution from the integrating factor method

Given
Assume $$a\left( x \right)\ne 0$$ for all $$x$$, rearranging equation 7.1 gives

 * {|style="width:100%" border="0"

$${y}'+\frac{a_{0}\left( x \right)}{a_{1}\left( x \right)}y=\frac{b\left( x \right)}{a_{1}\left( x \right)}$$
 *
$$\,(Eq.7.12)$$
 * }

Find
Find an expression for $$y\left( x \right)$$ in terms of $$a_{0}$$, $$a_{1}$$ and $$b$$

Solution
Define $${P}\left( x \right)=\frac{a_{0}\left( x \right)}{a_{1}\left( x \right)}$$ and $$Q\left( x \right)=\frac{b\left( x \right)}{a_{1}\left( x \right)}$$ Equation 7.12 is then in the form $${y}'+{P}\left( x \right)y=Q\left( x \right)$$ The integrating factor for this form is $$I\left( x \right)=\exp \left[ \int{ {P}\left( x \right)}dx \right]$$ Introducing a dummy factor $$t$$ and substituting in for $$P\left( x \right)$$

 * {|style="width:100%" border="0"

$$\exp \left[ \int\limits_ – ^{x}{\frac{a_{0}\left( t \right)}{a_{1}\left( t \right)}dt} \right]\frac{dy}{dx}+\exp \left[ \int\limits_ – ^{x}{\frac{a_{0}\left( t \right)}{a_{1}\left( t \right)}dt} \right]\frac{a_{0}\left( x \right)}{a_{1}\left( x \right)}y=\frac{b\left( x \right)}{a_{1}\left( t \right)}\exp \left[ \int\limits_ – ^{x}{\frac{a_{0}\left( t \right)}{a_{1}\left( t \right)}dt} \right]$$ Reducing
 *
$$\,(Eq.7.13)$$
 * }
 * {|style="width:100%" border="0"

$$\frac{d}{dx}\left[ \exp \left[ \int\limits_ – ^{x}{\frac{a_{0}\left( t \right)}{a_{1}\left( t \right)}dt} \right]y \right]=\frac{b\left( x \right)}{a_{1}\left( x \right)}\exp \left[ \int\limits_ – ^{x}{\frac{a_{0}\left( t \right)}{a_{1}\left( t \right)}dt} \right]$$ Integrating
 *
$$\,(Eq.7.14)$$
 * }

$$y=A\exp \left[ -\int\limits_ – ^{x}{\frac{a_{0}\left( t \right)}{a_{1}\left( t \right)}dt} \right]+\exp \left[ -\int\limits_ – ^{x}{\frac{a_{0}\left( t \right)}{a_{1}\left( t \right)}dt} \right]\int\limits_ – ^{x}{\frac{b\left( s \right)}{a_{1}\left( s \right)}ds\exp }\left[ -\int\limits_ – ^{s}{\frac{a_{0}\left( t \right)}{a_{1}\left( t \right)}dt} \right]ds$$
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 *
$$\,(Eq.7.15)$$
 * }

Part 3
Solution to linear, first order, ordinary differential equation with varying coefficients using Euler integrating factor method.

Given
Given an equation in the form of 7.1, where $$a_{1}\left( x \right)=x^{2}+1$$, $$a_{0}\left( x \right)=x$$, and $$b\left( x \right)=2x$$, equation 7.1 becomes
 * {|style="width:100%" border="0"

$$\left( x^{2}+1 \right){y}'+xy=2x$$ Rearranging
 *
$$\,(Eq.7.16)$$
 * }
 * {|style="width:100%" border="0"

$${y}'+\frac{x}{x^{2}+1}y=\frac{2x}{x^{2}+1}$$ Equation 7.17 is in the form $${y}'+{P}\left( x \right)y=Q\left( x \right)$$ The integrating factor for this form is $$I\left( x \right)=\exp \left[ \int{ {P}\left( x \right)}dx \right]$$ or
 *
$$\,(Eq.7.17)$$
 * }

 * {|style="width:100%" border="0"

$$I\left( x \right)=\exp \left[ \int{\frac{x}{x^{2}+1}}dx \right]=\exp \left[ \frac{1}{2}\ln \left( x^{2}+1 \right) \right]=\sqrt{x^{2}+1}$$||
$$\,(Eq.7.18)$$ Multiply 7.17 by 7.18 and simplify
 * }
 * {|style="width:100%" border="0"

$$\frac{d}{dx}\left( y\sqrt{x^{2}+1} \right)=\frac{2y}{\sqrt{x^{2}+1}}$$ Integrating
 *
$$\,(Eq.7.19)$$
 * }
 * {|style="width:100%" border="0"

$$\int\limits_ – ^{x}{\frac{d}{dx}\left( y\sqrt{x^{2}+1} \right)}=\int\limits_ – ^{x}{\frac{2y}{\sqrt{x^{2}+1}}dy}$$ Gives
 *
$$\,(Eq.7.20)$$
 * }

$$\,y=2$$
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 * style="width:2%; padding:1px; border:2px solid #0000FF" |
 *
$$\,(Eq.7.21)$$
 * }

Contributing Members

 * Solved by Kurt Schulze
 * Posted by James Roark 12:32, 18 September 2010 (UTC)
 * Proof read by Adam Franklin 20:23, 20 September 2010 (UTC)
 * Proof read by Egm6321.f10.team03.sigillo 20:45, 21 September 2010 (UTC)

=Problem 8=

Given
From $$\displaystyle p. 10-3$$ of the lecture notes there are two integration constants

$${k}_{1}$$ is contained in $$\displaystyle (1) p. 10-3 $$

 * {| style="width:100%" border="0"

h \left( x\right)= \exp \left[ \int^{x} {a}_{0} \left( s\right)ds\right] $$ $$
 *
$$\displaystyle (Eq 8.1)
 * style= |
 * }

$$ {k}_{2}$$ is contained in $$\displaystyle (3) p. 10-3 $$

 * {| style="width:100%" border="0"

\int^{x} h\left( s\right)b\left( s\right)ds $$ $$
 *
$$\displaystyle (Eq 8.2)
 * style= |
 * }

Find
Show that $$ {k}_{1}$$ is not necessary

Solution
To show this we will use $$\displaystyle (6) p. 10-3$$ and show that $$ {k}_{1}$$ cancels out of the equation, but first we must find the constants $$ {k}_{1}$$ and $$ {k}_{2}$$ from equations $$\displaystyle (Eq 8.1) $$ and $$\displaystyle (Eq 8.2) $$

 * {| style="width:100%" border="0"

y \left( x\right)= \frac{1}{h \left( x\right)} \int^{x}h \left(s \right)b \left( s\right)ds $$ $$
 *
$$\displaystyle (6) p. 10-3
 * style= |
 * }

When we perform the integration in the RHS of $$\displaystyle (Eq 8.1) $$ it becomes

 * {| style="width:100%" border="0"

\exp \left[ \int^{x} {a}_{0} \left( s\right)ds\right]= \exp \left[  \int {a}_{0} \left( x\right)dx + k \right] $$ $$
 *
$$\displaystyle (Eq 8.3)
 * style= |
 * }

Using the properties for exponential functions $$ \exp \left[ a+b\right] = \exp \left[ a\right]\exp \left[b\right]$$

Now $$\displaystyle (Eq 8.3)$$ becomes

 * {| style="width:100%" border="0"

\exp \left[ \int {a}_{0} \left( x\right)dx \right]\underbrace{\exp \left[k\right]}_{=: k_1} $$ $$
 *
$$\displaystyle (Eq 8.4)
 * style= |
 * }

Plugging this back into $$\displaystyle (Eq 8.1) $$ gives us
 * {| style="width:100%" border="0"

$$ h \left( x\right)= k_{1}\exp \left[ \int {a}_{0} \left( x\right)dx\right] $$ $$
 * style="width:25%; padding:1px; border:2px solid #0000FF" |
 * style="width:25%; padding:1px; border:2px solid #0000FF" |
 *
$$\displaystyle (Eq 8.5)
 * style= |
 * }

To find $$ {k}_{2}$$ we will evaluate the integral in $$\displaystyle (Eq 8.2) $$

 * {| style="width:100%" border="0"

$$ \int^{x} h\left( s\right)b\left( s\right)ds= \int h\left( x\right)b\left( x\right)dx + k_{2} $$ $$
 * style="width:25%; padding:1px; border:2px solid #0000FF" |
 * style="width:25%; padding:1px; border:2px solid #0000FF" |
 *
$$\displaystyle (Eq 8.6)
 * style= |
 * }

Finally plug equations $$\displaystyle (Eq 8.5) $$ and $$\displaystyle (Eq 8.6) $$ into $$\displaystyle (6) p. 10-3 $$

 * {| style="width:100%" border="0"

y \left( x\right)= \frac{1}{\cancelto{}{k_{1}}\exp \left[  \int {a}_{0} \left( x\right)dx\right]}\left[ \int \cancelto{}{k_{1}}\exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +k_2\right] $$ $$
 *
$$\displaystyle (Eq 8.7)
 * style= |
 * }

This shows $$k_1$$ is not necessary
 * {| style="width:100%" border="0"

$$ y \left( x\right)= \frac{1}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]}\left[ \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +k_2\right] $$ $$
 * style="width:25%; padding:1px; border:2px solid #0000FF" |
 * style="width:25%; padding:1px; border:2px solid #0000FF" |
 *
$$\displaystyle (Eq 8.8)
 * style= |
 * }

Given
From the meeting 10 lecture notes

 * {| style="width:100%" border="0"

y \left( x\right)= \frac{1}{h \left( x\right)} \int^{x}h \left(s \right)b \left( s\right)ds $$ $$
 *
$$\displaystyle (6) p. 10-3
 * style= |
 * }

Similarly from King et al, p. 512

 * {| style="width:100%" border="0"

y = A\exp \left\{  - \int ^{x}{P} \left( t\right)dt\right\}+\exp \left\{  - \int ^{x}{P} \left( t\right)dt\right\}\int ^{x}{Q} \left( s\right)\exp \left\{  \int ^{x}{P} \left(t\right)dt\right\}ds $$ $$
 *
$$\displaystyle K p. 512
 * style= |
 * }

Find
Show that $$\displaystyle (6) p. 10-3 $$ agrees with $$\displaystyle K p. 512 $$, i.e.

 * {| style="width:100%" border="0"

y = A{y}_{H}\left(x \right)+{y}_{P}\left(x \right) $$ $$
 *
$$\displaystyle (Eq 8.9)
 * style= |
 * }

Solution
To show that thes two equations agree we need to get them in the form of $$\displaystyle (Eq 8.9)$$. Fortunately, $$\displaystyle K p. 512 $$ is already in this form where

 * {| style="width:100%" border="0"

A{y}_{H}\left(x \right)= A\exp \left\{ - \int ^{x}{P} \left( t\right)dt\right\} $$ $$
 *
$$\displaystyle (Eq 8.10)
 * style= |
 * }

 * {| style="width:100%" border="0"

{y}_{P}\left(x \right)= \exp \left\{ - \int ^{x}{P} \left( t\right)dt\right\}\int ^{x}{Q} \left( s\right)\exp \left\{  \int ^{x}{P} \left(t\right)dt\right\}ds $$ $$
 *
$$\displaystyle (Eq 8.11)
 * style= |
 * }

Now, using $$\displaystyle (Eq 8.8) $$ which is the expanded form of $$\displaystyle (6) p. 10-3 $$ we can further expand and rearrange to get

 * {| style="width:100%" border="0"

y \left( x\right)= \frac{1}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]} \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +\frac{k_2}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]} $$ $$
 *
$$\displaystyle (Eq 8.12)
 * style= |
 * }

 * {| style="width:100%" border="0"

y \left( x\right)= \exp \left[  -\int {a}_{0} \left( x\right)dx\right] \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +k_2\exp \left[  -\int {a}_{0} \left( x\right)dx\right] $$ $$
 *
$$\displaystyle (Eq 8.13)
 * style= |
 * }

Now we have $$\displaystyle (6) p. 10-3 $$ in the form of $$\displaystyle (Eq 8.9)$$ where

 * {| style="width:100%" border="0"

A{y}_{H}\left(x \right)= k_2\exp \left[ -\int {a}_{0} \left( x\right)dx\right] $$ $$
 *
$$\displaystyle (Eq 8.14)
 * style= |
 * }

 * {| style="width:100%" border="0"

{y}_{P}\left(x \right)= \exp \left[ -\int {a}_{0} \left( x\right)dx\right] \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx $$ $$
 *
$$\displaystyle (Eq 8.15)
 * style= |
 * }

By comparing $$\displaystyle (Eq 8.10) $$ with $$\displaystyle (Eq 8.14)$$ and $$\displaystyle (Eq 8.11) $$ with $$\displaystyle (Eq 8.15) $$ we can see that these equations are in agreement

Given

 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

$$
 * $$\displaystyle {{y}^{'}}+{{a}_{0}}y=0$$
 *
$$\displaystyle (Eq 8.16)
 *
$$\displaystyle (Eq 8.16)
 * }
 * }

Find

 * {| style="width:100%" border="0"

 * Find $$\displaystyle y_H$$
 * style= |
 * }
 * }

Solution
Since $$\displaystyle (Eq 8.16)$$ is the homogeneous portion $$\displaystyle (2) p. 10-3$$ and is seperable, we can solve directly for $$ y_H$$

 * {| style="width:100%" border="0"

\frac{1}{y}{y}^{'} = -{a}_{0} $$ $$
 *
$$\displaystyle (Eq 8.17)
 * style= |
 * }

Integration of this equation gives

 * {| style="width:100%" border="0"

\ln \left( y\right)= -\int^{x} {a}_{0} \left( s\right)ds + C $$ $$
 *
$$\displaystyle (Eq 8.18)
 * style= |
 * }

Solving for $$y$$ gives

 * {| style="width:100%" border="0"

y= \exp\left[ -\int^{x} {a}_{0} \left( s\right)ds + C \right]= A\exp\left[ -\int^{x} {a}_{0} \left( s\right)ds\right]= y_H $$ $$
 *
$$\displaystyle (Eq 8.19)
 * style= |
 * }

Contributing Members
Solved and Posted by:--Egm6321.f10.team3.cook 23:47, 19 September 2010 (UTC)
 * Solved and Proof read by Adam Franklin 20:23, 20 September 2010 (UTC)

=Problem 9- Find N1-ODE that is either exact or can be made exact=

Given
Class of N1-ODE is given by (eq. (1),(2)& (3), page 12-1 of the lecture note):

 * {| style="width:100%" border="0"

$$ \underbrace{\bar{b} \left( x\right)  c\left( y\right)}_{N\left( x,y \right)}{y'} +   \underbrace{a\left( x\right) \bar{c}\left( x\right)}_{M\left( x,y \right)} = 0 $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\,(Eq.9.1)$$
 * }

where
 * {| style="width:100%" border="0"

$$ \bar{b} \left( x\right) = \int\limits_ – ^{x}{{b} \left( s\right)}ds $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\,(Eq.9.2)$$
 * }

 * {| style="width:100%" border="0"

$$ \bar{c} \left( y\right) = \int\limits_ – ^{y}{{c} \left( s\right)}ds $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\,(Eq.9.3)$$
 * }

Given values
 * {| style="width:100%" border="0"

$$ a\left( x\right) = sin(x^3) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\displaystyle
 * }

 * {| style="width:100%" border="0"

$$ b\left( x\right) = cos(x) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\displaystyle
 * }

 * {| style="width:100%" border="0"

$$ c\left( y\right) = e^{2y} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\displaystyle
 * }

Find

 * (1) Find a N1-ODE that is either exact or can be made exact

 * {| style="width:100%" border="0"

\phi(x,y)=k $$ $$
 * (2) Find, $$\displaystyle
 * (2) Find, $$\displaystyle
 *
$$\displaystyle
 * }

Solution
Part 1
 * {| style="width:100%" border="0"

$$ \bar{b} \left( x\right) = \int\limits_ – ^{x}{{b} \left( s\right)}ds = \int\limits_ – ^{x}{cos(s)}ds = sin(x) $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\,(Eq.9.4)$$

 * }
 * {| style="width:100%" border="0"

$$ \bar{c} \left( y\right) = \int\limits_ – ^{y}{{c} \left( s\right)}ds = \int\limits_ – ^{y}{e^{2s}} ds = \frac{1}{2}e^{2y} $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\,(Eq.9.5)$$
 * }

Now Eq.1 becomes:
 * {|style="width:100%" border="0"

$$ sin(x) e^{2y} y' + \frac{1}{2}sin(x^{3}) e^{(2y)} = 0 $$
 *
$$\,(Eq.9.6)$$
 * }

Eq. 9.6 is the form of
 * {| style="width:100%" border="0"

$$\,M(x,y)+N(x,y)y'=0$$ and first condition of Exactness is satisfied. $$
 *
$$\,(Eq.9.7)
 * style= |
 * }

The condition below is not only necessary but also sufficient for Eq.9.6 to be an exact differencial equation.

 * {| style="width:100%" border="0"

$$ $$
 * $$\frac{\partial M }{\partial y} = \frac{\partial N }{\partial x}
 * $$\frac{\partial M }{\partial y} = \frac{\partial N }{\partial x}
 *
$$\displaystyle (Eq.9.8)
 * }
 * }

 * {| style="width:100%" border="0"

 * $$\frac{\partial M }{\partial y} = \frac{\partial (\frac{1}{2}sin(x^{3})e^{2y})}{\partial y} =sin(x^{3})e^{2y} $$
 *
$$\,(Eq.9.9)$$
 * }
 * }
 * {| style="width:100%" border="0"

 * $$\frac{\partial N }{\partial x} = \frac{\partial (sin(x)e^{2y})}{\partial x} = cos(x)e^{2y}$$
 *
$$\,(Eq.9.10)$$
 * }
 * }

Eq.9.9 & 9.10 reveals that

 * {| style="width:100%" border="0"

$$
 * $$\frac{\partial M }{\partial y}  \neq  \frac{\partial N }{\partial x} $$
 *
$$\displaystyle
 *
$$\displaystyle
 * }
 * }

Hence the Eq.9.6 is not exact.

But it can be made exact by using Integrating factor method.Refer to Eq(7), page 10-1 of the lecture notes, and let's multyply Eq.(9-6) by h(x) in order to have an exact differential equation.

 * {| style="width:100%" border="0"

h(x) = exp[ \int^{x} {(-\frac {1}{N}(N_{x} -M_{y}))dx]} $$ $$
 *
$$\displaystyle
 * style= |
 * }
 * We can rewrite Eq.9.6 by cancelling $$\displaystyle e^{2y}$$ from both term as:
 * {| style="width:100%" border="0"

$$\underbrace{sin(x)}_{N(x,y)}{y}'+\underbrace{\frac{1}{2}sin({{x}^{3}})}_{M(x,y)}=0$$ $$ Now:
 *
$$\displaystyle
 * style= |
 * }
 * {| style="width:100%" border="0"

h(x) = exp[ \int^{x} {(-\frac {1}{sin(x)}(cos(x) - 0))dx]} $$ $$
 *
$$\displaystyle
 * style= |
 * }

 * {| style="width:100%" border="0"

= exp[ \int^{x} {(-\frac {cos(x)}{sin(x)})dx]} $$ $$
 *
$$\displaystyle
 * style= |
 * }

 * {| style="width:100%" border="0"

= exp[ -ln{(sin(x))}] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle
 * style= |
 * }

 * {| style="width:100%" border="0"

= \frac {1}{sin(x)}$$
 *
$$\,(Eq.9.11)$$
 * }

Now Exact differential equation is given by
 * {|style="width:100%" border="0"

$$ That is
 * $$h(x)[sin(x) e^{2y} y' + \frac{1}{2}sin(x^{3}) e^{(2y)}] = 0
 * $$h(x)[sin(x) e^{2y} y' + \frac{1}{2}sin(x^{3}) e^{(2y)}] = 0
 * style= |
 * }

 * {| style="width:100%" border="0"

$$sin(x)\frac{1}{sin(x)}y' + \frac{1}{2}sin(x^3)\frac{1}{sin(x)} = 0$$
 * style= |
 * }

 * {| style="width:100%" border="0"

$$ y' + \frac{1}{2} \frac{sin(x^3)}{sin(x)}= 0$$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 *
$$\,(Eq.9.12)$$
 * }

Part 2

 * Find $$\displaystyle\phi(x,y)=k

$$
 * We know that
 * {| style="width:100%" border="0"

$$\,M(x,y)=\phi_{x}{(x,y)}$$ $$
 *
$$\,
 *
$$\,(Eq.9.13)$$
 * }

 * {| style="width:100%" border="0"

$$\,N(x,y)=\phi_{y}{(x,y)}$$
 *
$$\,(Eq.9.14)$$
 * }

 * Also,
 * {| style="width:100%" border="0"

$$\,\phi_{x}{(x,y)} = \frac{\partial \phi{(x,y)}}{\partial {x}}$$
 *
$$\,(Eq.9.15)$$
 * }
 * {| style="width:100%" border="0"

$$\,\phi_{y}{(x,y)} = \frac{\partial \phi{(x,y)}}{\partial {y}dx}$$
 *
$$\,(Eq.9.16)$$
 * }

By integrating Eq.9.15 and 9.16 with respect to x, y, respectively, we can have equations as given below.

 * {| style="width:100%" border="0"

$$\,\int \phi_{x}{(x,y)}dx = \phi {(x,y)} = \int M(x,y) dx + k(y)$$

 *
$$\,(Eq.9.17)$$
 * }

 * {| style="width:100%" border="0"

$$\,\int \phi_{y}{(x,y)}dy = \phi {(x,y)} = \int N(x,y) dy + k(x)$$
 *
$$\,(Eq.9.18)$$
 * }

Now we can calculate $$\displaystyle k(y)$$ by differentiating Eq.9.17 with respect to y and equating to Eq.9.14=>

 * {| style="width:100%" border="0"

$${{k}^{'}}\left( y \right)=N\left( x,y \right)=1$$ $$
 *
$$\,
 * style= |
 * }
 * Now integrate the above equation =>

 * {| style="width:100%" border="0"

$$\displaystyle k(y)=y+k1$$ $$ Then $$\displaystyle\phi(x,y)$$ is obtained by substituting $$\displaystyle k(y)$$ in Eq.9.17:
 *
$$\,
 * style= |
 * }
 * {| style="width:100%" border="0"

$$\phi \left( x,y \right)=\int{M\left( x,y \right)dx+y+k1}$$ $$
 *
$$\,
 * style= |
 * }

That is
 * {| style="width:100%" border="0"

$$\phi \left( x,y \right)=\frac{1}{2}\int{\frac{\sin \left( {{x}^{3}} \right)}{\sin x}dx+y=k}$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 *
$$\,
 * style= |
 * }

Contributing Members

 * Solved Part 2 and proof read by-Egm6321.f10.team3.Sudheesh 04:24, 22 September 2010 (UTC)
 * Solved and posted by-Egm6321.f10.team3.Hong SJ 22:30, 21 September 2010 (UTC)

=Contributing Team Members=
 * Adam Franklin 18:56, 12 September 2010 (UTC) solved 1 & 8 and proofed 1, 2 , 7 & 8
 * Sudheesh 18:11, 14 September 2010 (UTC), solved 1 & 4, posted 4 and proofed 1 & 5
 * Jimmy Roark 14:26, 9 September 2010 (UTC)
 * Michele Sigilló 22:07, 12 September 2010 (UTC), solved 1 & 5, posted 1,5 and 6, proof read 2,3,4,6
 * Kurt Schulze 01:10, 13 September 2010 (UTC), solved 6 & 7, proofed 2 & 4
 * Chris Cook 11:06, 14 September 2010 (UTC)
 * Hong SJ 11:12, 14 September 2010 (UTC)

=Notes and references=

title: User:Egm6321.f10.team03/Hwk2


text:
User talk:Manipande

 Hello Manipande, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 04:37, 15 September 2010 (UTC)

title: User talk:Manipande


text:
User:PersonalGenotypingExperiment/Very Rough Draft

1. Listing symptoms, disease and traits
Primarily complaint is mood disorders, dysregulation of mood and arousal.

 * High IQ
 * Anxiety
 * Depression
 * Circadian Rhythm disturbance
 * Cushingoid features

2. Family History
Conditions are clearly familial (but familial doesn't automatically imply genetic).
 * High IQ-- both parents
 * Anxiety-- mother, mothers parents and siblings
 * Depression-- both parents, mother's father
 * Circadian Rhythm disturbance - father only

Rough Summary of what to look for
In the 23andme data, I will be most interest in SNPs in genes:
 * have been linked in published literature to my symptoms or my diagnoses
 * encode for signaling molecules or receptors, especially those tied to mood or arousal
 * implicated in the interregulation of limbic, hypothalamic, pituitary, adrenals, gonads system
 * implicated in regulation of neurogenesis
 * implicated in epigenetic regulation of other implicated genes.

This is a very rough draft leaving out much, but it documents the rough search strategy i'll employ.


text:
Motivation and emotion/Textbook/Motivation/Unconscious motivation

Overview
This chapter will discuss the theories, concepts, and ideas behind unconscious motivation. Firstly it gives a short description of human motivation after which it will then consider the idea behind unconscious motivation. The second part of this chapter will briefly outline the history behind the unconscious motivation concept, after which it will then examine Freud’s psychodynamic perspective, including his theory on dreams and dream analysis. From there this chapter will discuss post Freudian Evolution and consider Carl Jung's theories of the unconscious. The final section of this chapter focuses on additional theories in motivation and unconsciousness

Hidden and unknown desires (What is Unconscious motivation)

Motivation is the desire to achieve a goal, it can be defined as the process which initiates, guides, and maintains goal adjusted behaviour (Ouellette, 1998). Motivation is not just be self- proclaimed goals but also included specific needs such as eating, drinking, and sleep (Ouellette, 1998). According to Ouellette, (1998) there are four basic principles of motivation.
 * our motives are obtained through learning
 * motives can change
 * motives come from within and without (e.g. motives for hunger come from within and motives such as desire are from without)
 * different behaviours can come from the same motive (e.g a child may require attention from her parents so she behaves badly in order to gain their affection). Motives are generally categorized as physiological, social, and all those in between (Ouellette, 1998).

To understand unconscious motivations we must first consider the workings of the unconscious mind. Some researchers today would characterize the brain as a computer, programmed by something that isn't connected to itself, but rather a part of its self, refered to as the unconscious mind. The unconscious mind is not physical, therefore, it can not been seen nor measured scientifically (Archard, 1984). The questions than is, are humans ruled by this non physical unconscious mind? According to Archard, (1984) they are not. He considered that human behaviour is simply stimulus response mechanisms. Therefore, the mind is a mechanism and individuals can be programmed (Archard, 1984). According to Bargh, Gollwitzer,  Lee- Chai, Barndollar, & Trotschel, (2001) individuals are motivated to make choices based only on their conscious thoughts. According to, Archard, (1984) & Freud, (1961/2004) a large amount of human behaviour is stimulated by unconscious motives (e.g. Sigmund Freud believed that the majority of all human behaviour is a result of their desires, impulses, and memories that had been repressed into an unconscious state). According to Maslow, the average person is more often unconscious than conscious. He believed unconscious motives take central roles in determining the way in which people behave (Archard, 1984).

It is very difficult to defined unconscious motivation but in essence it is that of which we are unaware. It is the unconscious desires, instincts, and needs of humans. Some researchers believe that our unconscious motivaties only get acted on when we are stressed and anxious (Archard, 1984; Freud, 1961/2004; Reason, 2000;). While other researchers believe we have total control over them (Weston, 1999).

The history behind the unconscious motivation concept
The German philosopher Ernst Platner introduced the term ‘unconscious’ around the 18th century, however the significance of what we now call unconscious processes were recognized by many earlier influential thinkers such as Nietzsche, Shakespeare, Da Vinci, Paracelcus, Dante, and St Thomas Aquinas (Archard, 1984: Jacobs, 2003). It may well be that the first written reference to the likelihood of unconscious psychological processes were made by the Greek philosopher Plotinus in the third century AD, who remarked that “the absence of a conscious perception is proof of the absence of mental activity” (Koestler, 1964, p. 148).

The notion of an unconscious has been the source of much controversy for more than a century. Two main opposing views, have taken on various developments over the years and challenged each other down till today. During much of the later half of the 19th century, in clear contradiction to the exploratory position taken by the above philosophers (in favour of an unconscious mind), William James, mocked the notion of an unconscious, marshalling many arguments against it (Freud,1961/2004). The second view was Freud's theory of unconscious motivation. Not long after the turn of the century, behaviourism took the position not only against an unconscious, but also against consciousness (Weston, 1999). During this time Freud’s psychoanalytical prospective remained an advocate and defender of the unconscious mind.

Closer to our time, the debate is studied at a more empirical level, with the introduction of subliminal methods (Weston, 1999). However, many cognitive psychologists have drawn the line against accepting a dynamic unconscious, involving, affecting, and motivation humans (Bargh & Trotschel, 2001; Weston, 1999). In modern day cognitive psychology, researchers have endeavoured to strip the notion of the unconscious from its Freudian traditions. Today cognitive researcher use alternative terms such as, "implicit" and "explicit" to explain the inner and outer workings of human behaviour (Weston, 1999). Cognitive research methods of unconscious processes are based on fewer theoretical assumptions and are more empirically driven, they also emphasize the extent to which cognitive processes occur outside of awareness (Weston, 1999).

Sigmund Freud's theories of the unconscious

Freud's theory of the unconscious is immensely complex. The information that follows is a severely condensed account of his work. It focuses on those parts of his theory that are related to the unconscious mind and unconscious motivations, purely for the benefit of these chapters.

Freud believed that the psyche had unconscious ways of interacting with the world that were totally different from those of consciousness (Francher, 1973). Like consciousness, the unconscious had desires, intentions, aims, wishes, and thoughts, but rather than operating in an open and logical manner, they were purely instinctual (Francher, 1973). The unconscious was many things to Freud; a realm of repressed wishes, the carrier and generator of childish fantasies, and especially sexual fantasies. But first and foremost, it was the cauldron of the instincts, the driving forces for all human activity and behaviour (Fancher, 1973).

Freud believed that there were different levels for unconsciousness and consciousness. First he believed in the level of conscious thought, (i.e.) the material that we are actively aware of at any given time (Francher, 1973). Secondly, he believed in the pre-conscious mind. This consisted of thoughts that were unconscious at any one moment, but could be easily recalled into consciousness if we so needed them (Eder, 2009). For example, the pin code for our ATM cards. The final level was the unconscious mind. For Freud, this consisted of thoughts, memories, urges, feelings, and fantasies that we were unaware of because they were actively kept in the unconscious due to their unacceptable nature (Freud, 1961/2004). Freud argued that people repressed issues and obsessions such as sexual urges, aggression, and memories that produced anxiety, into their unconscious. He saw repression as an active continuous process (Freud, 1961/2004).

Although Freud described three levels of thought, he did not give a clear cut separation between conscious, preconscious, and unconscious thought; rather, he suggested different levels within each (Eder, 2009). For example, a person’s repressed material may weaken at times, so that previously unconscious material becomes conscious (Eder, 2009). He believed that this unconscious material was in a modified form. For example: symptoms of psychological disturbance or illness, the emergence of unfamiliar impulses under the influence of drugs, or as dreams (Eder, 2009), (for more information on dreams please see heading, Freud and Dream Analysis). It is important to note that there has been no scientific evidence, even now, that confirms or that denies Freud's notions of the unconscious, preconscious, or conscious mind. Freud's theoies are based on work with his own patients and not from empirical evidence. Within his work he addresses a wide range of phenomena. However, he often revisited his ideas, which makes his work extremely difficult to follow.

Freud's defence mechanisms

 * Repression http://en.wikipedia.org/wiki/Psychological_repression
 * Denial http://en.wikipedia.org/wiki/Denial
 * Projection http://en.wikipedia.org/wiki/Psychological_projection
 * Reaction formation http://en.wikipedia.org/wiki/Reaction_formation
 * Rationalisation http://www.thefreedictionary.com/rationalisation
 * Conversion Reaction http://en.wikipedia.org/wiki/Conversion_disorder
 * Phobic Avoidance
 * Displacement http://en.wikipedia.org/wiki/Displacement_(psychology)
 * Isolation http://en.wikipedia.org/wiki/Isolation
 * Undoing http://en.wikipedia.org/wiki/Undoing_(psychology)

For more information on Freud's theories of the unconscious please click this link http://www.youtube.com/watch?v=Nm7XGiFhKeE

Freud and dream analysis
The notion of a unconscious mind was essential to Sigmund Freud's theories and assisted him in finding meaning within dreams. For Freud, the fact that people dream at all was taken as an example of the activity in the unconscious mind (Freud, 1961/2004). Noting the content of dreams gave Freud clues about how processes in the unconscious mind operated, as well as methods to uncover the repressed information from within it (Freud, 1961/2004; Eder, 2009). He believed that dreams were wish fulfilling, meaning that in our dreams we act out unconscious desires (Eder, 2009). Freud’s dream analysis emphasized two levels of dream content. First the manifest content which is the description of the dream as recalled by the dreamer (Eder, 2009). For example if you dreamed about coming to school naked, the manifest content is your nudity, the school, the other people around you, and so on. The second was the latent content, which is the unconscious meaning of the manifest (literal) content (Eder, 2009). Freud considered that even during sleep, our egos protected us from the material in our unconscious minds by presenting our repressed desires in the forms of symbols. Therefore, coming to school naked would represent a symbol for some kind of repressesd matter (Eder, 2009).

In more detail, Freud felt that the manifest content was not a true portrayal of the unconscious mind. He believed that the dreamer unconsciously censors the information or uses symbols to represent key elements to avoid becoming too disturbed by the dream (Eder, 2009; Freud, 1961/2004). He felt that skilled interpretation was often necessary to get to the real meaning of a dream. Therefore, the task of analysis was used to identify the latent content of dreams (Freud, 1961/2004). He suggested that much of the unconscious content in dreams were sexual in nature (Freud,1961/2004). And that most symbols of dreams had a personal meaning for the dreamer and even identified many commonly occurring dream symbols. For example he suggested that snakes or knives represented the penis; a ladder or staircase, sexual intercourse; baldness or tooth extraction, castration fears (Freud, 1961/2004).

Freud believed that different styles of thinking were associated with different levels of consciousness (Francher, 1973). Dreams for example, represented what he refferred to as primary process thinking (Freud, 1961/ 2004). For Freud, dreams demonstrate the activity by the way in which events are often oblivious to the categories of time and space or extreme contradiction (Jacobs, 2003). Primary process thinking differs from conscious thinking in two ways. First, it is under the influence of the pleasure principle rather than the reality principle, and therefore, will choose objects according to what it thinks will give the most pleasure without regard to the availability or appropriateness of these objects in reality (Jacobs, 2003). The infantile mind, the primitive mid, and the existence of dreams are all aspects of the primary processes (Jacobs, 2003). The pleasure principle is the urge to have our desires met. This is not an active urge to seek pleasure, but rather an instinct or impulse to avoid displeasure, distress, or pain (Eder, 2009). It is about preserving a balance within.

Secondary process thinking is under the surveillance of the reality principles (Francher, 1973). Their aim is also to produce pleasure, or more commonly, reduce displeasure (Francher, 1973). The secondary processes are governed by reason and follow the pattern of logical thinking, they recognise temporal and spatial relationships and represent the function of the ego (Francher, 1973). Secondary process thinking is characteristic of conscious and preconscious thought. Freud (1961,2004) suggests, that the pleasure principle is a primitive, innate instinct that drives our behaviour while the reality principle is learnt as we develop.

For Freud dreams were wish fulfilling, he proposed that complex dreams are produced when an event during the day, typically an insignificant event, has a likeness to a childhood memory and is connected to a dangerous, previously repressed wish (Freud, 1961/2004). During the day an individual can keep this wish from consciousness, but when they relax at night, the wish can make its way into experience but only if it has been adequately concealed (Freud, 1961/2004). He adds that it is the function of ‘dream work’ that protects the sleeper from waking by transforming the wish through condensation displacement, dramatization, and symbolization into a fantasy that both expresses and satisfies the wish (Freud, 1961.2004). However, it is disguised in such a way that it does not offend the ego and wake the sleeper.

Is there empirical validity for Freud’s dynamic unconscious
Research has examined subliminal perception in an attempt to provide evidence of a dynamic unconscious. Subliminal perception is thought to occur when a person receives stimuli without being consciously aware of it (Henk, Ruud, & Hans, 2008). Researchers use subliminal stimuli in a bid to show its affects on participants' behaviour, thus demonstrating the existence of unconscious motivational effects on behaviour. Henk & Hans, (2008) showed that participants expended more effort on a hand grip test when they had been subliminally primed than when they had not. They believed this demonstrate that mental processes in preparing and motivating behaviour happen outside a person's awareness (Henk & Hans, 2008).

Erdelyi, (1984) used subliminal words to measure participants' anxiety levels. When participants were given emotionally threatening words their anxiety levels intensified. However, when participants were shown non threatening words, anxiety levels stayed neutral. Erdelyi, (1984) suggests this may validate the notion that subliminal stimuli influence unconscious behaviour. (Erdelyi, 1984). To Reason, (2000) the unconscious is more about automatic mental processes rather than a dynamic unconscious. Evidence of dream content

Current research by Solms, (2000) on the neuropsychology of dreaming, demonstrates that may be an activation of desires and emotional mechanisms in the brain that initiate dreaming. He found that the manifest content of dreams were projected onto perceptual areas in the brain. He believed this to be evidence for Freud’s dream theory (Solms, 2000).

Post- Freudian Evolution of the Psychodynamic Approach

It has been very well documented that during Freud’s life time, several important figures in psychoanalysis who had been Freud’s students or close colleagues, were involved in disputes and consequently left his association for psychoanalysis. The most well know of these figures was Carl Jung, who was regarded as Freud’s chosen director to take over leadership of the psychoanalytic movement (Mitchell & Black 1995). The principles for their disagreements were centred on the nature of motivation (Mitchell & Black 1995).

Other well-known analysts who broke off from Freud included Ferenczi, Reich, Rank, and the popular Adler (Mitchell & Black 1995). These disagreements represented basic problems within the theoretical aspects of the psychodynamic approach. Three underlying questions were debated by Freud and his colleagues; what happens in the early years of life to produce later problems, what should therapist do to make psychoanalytic therapy better, and lastly how do unconscious processes and mechanisms operate (Mitchell & Black 1995). While Freud was still alive he dominated psychoanalysis, and those that disagreed with him were forced to set up separate institutes. After Freud’s death, it became possible to reopen the debate and many new concepts, theories, and approaches were reconsidered. These included:

 * Adler - developed a theory called individual psychology. He suggested that all humans suffer both psychological and social inferiority feelings, beginning at birth and continuing throughout childhood (Mitchell & Black 1995). To compensate for this, humans strive for superiority. If an individual is unable to do this they may end up with either an inferiority complex or a superiority complex (Mitchell & Black 1995). Unlike Freud's theories the neurotic personality is associated with the development of these inferiority or superiority complexes (Mitchell & Black 1995). Adler also modified Freud's treatment approaches to better suit the clients.

 * Jung - developed a model of the personality he called the psyche. It was a complex structure of opposing forces/ principles, which created the life- process energy that motives behaviour (Jung, 1964). Jung argued that behaviour was motivated by our future goals as well as by our past experiences (Jung, 1964). According to Jung, (1964) mental illness was caused by an imbalance in the psyche. To treat this he used word association, painting, and dream analysis to explore the patient’s unconscious. Jung believed in a collective unconscious (that lies deep within the psyche) and a personal unconscious (that is next to the ego) and contains all our personal experiences that have been blocked from awareness because they are unacceptable in nature. This part of his theory is simular to Freud’s theories of repression (Jung, 1964)

More Theories on Motivation and the Unconscious Mind

Maslow – Abraham Maslow described humans as having innate tendances towards healthy growth and development that he labelled instinctoid tendencies (Maslow, 1968). He believed these instinctoid tendencies were weak and easily over come by negative environmental influences. Maslow’s interests were in trying to understand what motivates us to go about our daily lives, and what makes us make the choices that we do (Maslow, 1968). From his observations, he described two distinct systems. The first were deficiency motives, which are basic needs that people are driven to fulfil (e.g. hunger, thirst, and the needs to be safe and loved) (Maslow, 1968). The second type of needs were growth motives, sometimes called 'being motives' or 'B-motives'. Maslow suggested that these were unique to each individual and gained intensity as they were being met. He believed these needs were about developing an individual’s potential (Maslow, 1968). There are two distinct differences between these concepts; deficiency motives are seen to ensure our survival, while growth needs represent a higher level of functioning that can result in a person becoming happier, healthier, and more fulfilled (Maslow, 1968).

Maslow’s hierarchy of needs

Maslow believed that needs were important for human survival and therefore he developed the notorious Hierarchy of Needs. This hierarchy consisted of five basic needs.

 * physiological needs – these include hunger, thirst, sleep, oxygen, sex, and the elimination of bodily waste. Once the physiological needs are satisfied we move on to the next step in the hierarchy (Malsow, 1968).

 * Safety Needs - these are the need for security, safety, and the need for law-abiding communities and a sense of order. Maslow alleged that the safety needs can clearly be observed in infants and young children when they are upset or scared. He also believed that if an infant does not get their security needs meet, that this will impact on the child’s development (Maslow, 1968).
 * Belongingness and Love Needs - Once our physiological and safety needs are meet, Maslow suggests that we long for belongingness and love (Maslow, 1968). Within the hierarchy Maslow defined two different types of love: D-love and B-love. D-love is the love we seek to meet the emptiness inside ourselves. Once our basic need for D-love has been meet than we move on to B-love, which is, the need for love and belongingness ??from?? ourselves (Maslow, 1968). (B here stands for “Being”, D for “Deficiency”)

 * The Esteem Needs- Esteem needs are the last of the basic needs and are divided into two types of needs. The first is based on our need to see ourselves as competent, achieving individuals. The second is the need for esteem based on the evaluation of others. This is the need for respect and admiration from other people around us (Maslow, 1968).

 * The Need for Self-Actualisation - The highest level of needs is for self-actualisation. Maslow believes once an individual's basic motivation needs are met then they start to focus on what they want from life. He suggests the needs of self-actualisers are qualitatively different; he describes them as metaneeds (Maslow, 1968). Metaneeds are, being concerned with higher aesthetic and moral values such as beauty, truth, justice, and ethics. Maslow also states, that not many individuals will achieve self-actualisation but many will strive to do so (Maslow, 1968).

Test your unconscious association {{Hide in print|
 * Drive Theory - This theory is based on the notion that humans are born with certain physiological needs. However if these needs are not being satisfied it creates a negative state in which the individual feels anxiety or tension (Zajonc, 1965). Within the Freudian psychodynamic approach drives theory referred to individual’s drives, motivations, instincts, and impulses that have clear intentions (Francher, 1973). Freud believed the psyche was torn between the Eros (which seeks wholeness through ever increasing unifications) and Thanatos (that wishes to achieve wholeness by breaking things apart until the absolutely simple is attained). Much of what happens in an individual's life, according to Freud, depends on which of these primal forces gains the upper hand (Francher, 1973).
 * Attribution Theory - An attribution is when an individual tries to make conclusions about another person’s behaviour. This theory is based on the notion that humans attribute internal and external factors which lead to increased motivation (Roesch & Amirkham, 1997). An internal attribution is any rationalization that occurs internally (or within) a person. For example, their mood, attitudes, personality, and abilities (Roesch & Amirkham, 1997). An external attribution is any explanation or rationalization that is located outside the person, such as the actions of others, social pressures, luck, or the nature of a situation (also known as situational attribution) (Roesch & Amirkham, 1997). The attributions people make are often driven by emotional and motivational drives (Roesch & Amirkham, 1997).


text:
User talk:Moofirst

 Hello Moofirst, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:18, 15 September 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:18, 15 September 2010 (UTC)

title: User talk:Moofirst


text: Greek Language/έχω

title: Greek Language/έχω