text:
Engineering Projects/Poppit/Howard Community College/Fall2011/501 boring

Copy the outline of the electronic sections of the team done information below:

Problem Statement
Identify if the matrix set-up is truly random in terms of both ratios of balloons as well as initial groupings of balloons.

Team Members
Put each team members name here with links to their personal page here. Real names do not have to be used.
 * thanson2045
 * hkang0839
 * Drew_Dollas

Summary
We counted the number of balloons of each color and came to the conclusion that the setup is random to an extent. The number of ballons of each color is random within a certain range (~15-45) and that once the number of balloons of each color is picked then the setup and placement of all of the balloons within the grid is truly random. team weekly reports.

Decision List
Which part of the project specifically we wanted to focus on and advance. We ended up deciding on exactly how random the setup of the game is.

Software List
Google Chrome

Time
Estimated 1-2 hours per person over the weekend. Actual time ended up being slightly over 2 hours per person.

Next Steps
Testing further and confirming our findings about how random the game is and also attempting to recreate a simpler version of the game that creates matrices of a similar nature.


text:
User:Drew Dollas

http://en.wikiversity.org/wiki/User:Drew_Dollas/enes100

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

title: User:Drew Dollas


text:
User:Ckpress/enes100/poppit

Link back to team EME

Write problem statement
In my process I used statistics to try and figure out how to beat poppit. Using the human average by playing a series of games and recording data then comparing with-in our team.

Assign Task1
Play poppit twenty times and record my average. Use the seven methods: 1. Left to right 2. Right to left 3. Top to bottom 4. Bottom to top 5. Chunks 6. Doubles 7. Color elimination

Compare actual work done to Task1
Team Averages

CFurby avg: 22.3 ARoberts avg: 20.6 Ckpress avg: 27.1

Week1 Narrative
I decided to use statistics to figure out the human average as and approach to how to reduce the left over balloons when playing poppit. I played a set of twenty plus games and record he/she average. I approached the task by using many different methods. First method was popping the ballons from left to right. Second method was from Right to left, top to bottom, bottom to top, then in chunks, doubles, and finally color elimination.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

title: User:Ckpress/enes100/poppit


text:
User:Thanson2045/enes100/poppit

Here I am going to put all information for fall 2011 Link back to team robots

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
Tasks are typically done over the weekend. They are documented during the week, presented during the week and the next weekends work is negotiated during the week. Week 0 is the week that task 1 is negotiated. Record your individual task here. Plan on working on it during the weekend between week0 and week1.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Thanson2045/enes100/poppit


text:
User:CFurby/enes100/poppit

Link back to team EME

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
In our process as a group we used statistics to try and figure out how to beat poppit. Using the human average by playing a series of games and recording data then comparing with-in our team. Hopefully, this helps us discover a way to create a way to pop all the balloons in Poppit every time the game is played.

Assign Task1
Week 1: I played the game Poppit 40 times over this weekend. My average number of left over balloons was 22.3. Once my teammates have substantial averages then I will calculate our group average.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:CFurby/enes100/poppit


text:
User:Drew Dollas/enes100/poppit

Link back to team robots

Give instructor top 3 project choice list
Just Poppit

Write problem statement
Identify if the matrix set-up is truly random in terms of both ratios of balloons as well as initial groupings of balloons.

Assign Task1
play 50 games of Poppit. write down the number of each color balloons for every game. Write down how many groups of each colored balloons there are for every game.

title: User:Drew Dollas/enes100/poppit


text:
User:Hkang0839/enes100/poppit

Here I am going to put all information for fall 2011 Link back to team robots

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
Tasks are typically done over the weekend. They are documented during the week, presented during the week and the next weekends work is negotiated during the week. Week 0 is the week that task 1 is negotiated. Record your individual task here. Plan on working on it during the weekend between week0 and week1.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Hkang0839/enes100/poppit


text:
User:Egm6321.f11.team4

=Report Page= Report #1

Report #2

Report #3

Report #4

Report #5

Report #6

Report #7

=Team Members= Junsu Shin

Yu Chen

Ankush Bhatia

Kexin Ren

Yongmin Chung

Sean Allen

=Class Page= EGM 6321 Class wiki

Fall 2011 Report table

Fall 2011 Lecture plan

title: User:Egm6321.f11.team4


text:
User:Cdiffendal7527/enes100

this is the new page for engineering projects

title: User:Cdiffendal7527/enes100


text:
User:Egm6321.f11.team4/HW1

= Problem 1.1 - First and Second Derivatives of the Equation of Maglev Trains =

Given
The equation of the meglev trains:$$\displaystyle f(Y^1 (t),t) $$

Find
Derive the first and second time derivatives for the given equation.

Partial Derivation of Multivariable Functions
Suppose $$\displaystyle u(x) $$ and $$\displaystyle v(x) $$ are functions of $$\displaystyle x $$. Then the first partial derivative with respect to $$\displaystyle x $$ of the multivariable function $$\displaystyle f(u,v)=f(u(x),v(x)) $$ is:

First Derivative
Now let $$\displaystyle s=Y^1(t) $$ for ease,so the first time derivative of the given function should be:

Use $$\displaystyle \dot Y^1(t) = \frac{dY^1}{dt} $$ for ease:

Second Derivative
Derive Eq. (1.3) with respect to $$\displaystyle t $$ again:

For ease, use $$\displaystyle f_{,x} := \frac{\partial f}{\partial x} $$ and $$\displaystyle f_{,xy} := \frac{\partial^2 f}{\partial x \partial y} $$, so (1.4) is actually:

Author
Contributed by Yu Chen

=Problem 1.2 - Derivation of the 1st and 2nd derivatives and its similarity with the Coriolis force= From [[media:Pea1.f11.mtg3.djvu|Mtg 3-2]]

Given
The equation of the meglev trains:

Find
Derive the first and the second derivatives of the Eq. (2.1) and show the similarity with the derivation of the Coriolis force.

Solution
To compute the first derivative of the function $$\displaystyle f(s,t) $$, the chain rule should be applied.

First Derivative
Substituting $$\displaystyle Y^1(t) $$ for $$\displaystyle s $$ in the Eq. (2-2).

Defining the notation as below.

Second Derivative
Taking a derivative of the Eq. (2.2) with respect to $$\displaystyle t $$.

Defining the notations below. Then, the Eq. (2.6) is redefined using the notations in the Eq. (2.7) as below. Substituting $$\displaystyle Y^1(t) $$ for $$\displaystyle s $$ in the Eq. (2.8).

Derivation for the Coriolis acceleration
Define $$\displaystyle \mathbf{r} $$ as the vector indicating the position of the origin of the reference frame $$\displaystyle A $$ on an object rotating with an angular velocity of $$ \displaystyle {}^{N}\boldsymbol{\omega}^{A} $$ with respect to inertial reference frame $$\displaystyle N $$ ( not accelerating observer ). Then, the velocity of the object as viewed by an observer fixed to the inertial reference frame $$\displaystyle N $$ is defined as below. Also, the acceleration of the object as viewed by an observer fixed to the inertial reference frame $$\displaystyle N $$ is defined as below. Substituting $$\displaystyle {}^{N} \mathbf{v} $$ from the Eq. (2.11) into the Eq. (2.12), Expanding the Eq. (2.13), In the Eq. (2.14), the term $$\displaystyle {}^{N}\boldsymbol{\alpha}^{A} $$ represents the angular acceleration of the object in the inertial reference frame. The term $$\displaystyle 2{}^{N}\boldsymbol{\omega}^{A} \times {}^{A}\mathbf{v} $$ in the last equation in the Eq. (2.14) is the so-called Coriolis acceleration.

Similarities
The Eq. (2-9) and the Eq. (2-14) are related as below.

Author
Contributed by Shin

=Problem 1.3 - Dimensional Analysis= From [[media:Pea1.f11.mtg3.djvu|Mtg 3-5]]

Given
Equation of motion (EOM) of wheel/magnet can be expressed as

thus, we have the $$\displaystyle {c_0(Y^1,t)}$$ term

Find
Analyze the dimension of all terms in the equation, and provide the physical meaning.

First term
in which so

Second term
in which so

Third term
in which so

Fourth term
in which so

Physical meaning
In conclusion, the external forces (F1,F2,T) as well as the inertia term Ma contribute to each element from the physical aspect.

Author
Contributed by Kexin Ren

=Problem 1.4 - Polar coordinate lines= From [[media:Pea1.f11.mtg4.djvu|Mtg 4-5]]

Given
A point in a 2-D plane

Find
Draw the polar coordinate lines emanating from the point

Author
Contributed by Kexin Ren

=Problem 1.5 - Simplification of Separated Equations= From [[media:Pea1.f11.mtg4.djvu|Mtg 4-5]]

Given: Separated Equations in curvilinear coordinates
A point $$\displaystyle \xi$$ in 3-D can be written in curvilinear coordinates as $$\displaystyle \xi = (\xi_1,\xi_2,\xi_3)$$

Applying separation of variables to a function, $$\displaystyle X(\xi) $$, i.e. $$\displaystyle X(\xi) = X_1(\xi_1)X_2(\xi_2)X_3(\xi_3)$$, we get following three decoupled 2nd order partial differential equations in $$\displaystyle X_i(\xi_i) $$,

Above, $$\displaystyle \xi_i$$ is the independent variable and $$\displaystyle X_i$$ is the dependent variable, and functions, $$\displaystyle g_i$$ and $$\displaystyle f_i$$ are dependent on $$\displaystyle \xi_i$$

Find: Simplified form of separated equations
Show that by changing the symbols to,

$$\displaystyle \xi_i \rightarrow x$$

$$\displaystyle X_i(\xi_i) \rightarrow y(x)$$

$$\displaystyle g_i(\xi_i) \rightarrow g(x)$$

$$\displaystyle f_i(\xi_i) \rightarrow a_0(x)$$

We can find following simplified form of the separated equations,

This is a 2nd order linear ordinary differential equation in y, similar to the general equation given in Differential Equations, King 2003, p.3

Solution:Simplification Process
By substituting the symbols the above equation becomes, Taking the derivative $$\frac{d}{dx}$$, appearing in the first term on LHS into the square brackets (in the above equation), and performing the differentiation we get,

Denoting the first order derivative as ' on the variables, and simplifying equations we get,

Now the terms can be rearranged to get the final desired equation,

Author
Contributed by Ankush Bhatia

=Problem 1.6 - Proof of Nonlinearity of Certain Function= From [[media:Pea1.f11.mtg5.djvu|Mtg 5-5]]

Given
The first term in EOM:

where

Find
Show function (6.1) is nonlinear with respect to $$\displaystyle Y^1 $$.

Definition of Nonlinearity
If a function $$\displaystyle F(x) $$ does NOT satisfy:

then it is nonlinear.

Proof of Nonlinerarity
Let $$\displaystyle F(x)=M[1-\bar{R} u_{,ss}^2 (x,t)] \ddot x $$, then:

If $$\displaystyle F(\alpha x + \beta y) = \alpha F(x) +\beta F(y) $$ is true, then:

is true for any real number $$\displaystyle \alpha, \beta $$ and any $$\displaystyle x, y $$ in the domain. That means:

is true for any real number $$\displaystyle \alpha, \beta $$ and any $$\displaystyle x, y $$ in the domain, which is absolutely not true. So equation (6.3) is not true for $$\displaystyle c_3(Y^1,t) \ddot Y^1 $$, and $$\displaystyle c_3(Y^1,t) \ddot Y^1 $$ is nonlinear.

Author
Contributed by Yu Chen

='''Problem 1.7 - Show that L2 is linear. '''= From [[media:Nm1.s11.mtg23.djvu|Mtg 6-3]]

Given
$$ L_2(\cdot)=\frac{d^2(\cdot)}{dx^2} + a_1\frac{d(\cdot)}{dx} + a_0(x)(\cdot) $$

Find
$$L_2(\cdot)$$ is linear.

Solution
If $$y_1, y_2$$ are the solution of $$(\cdot)$$ then, let $$(\cdot) = c_1y_1+c_2y_2$$

As a result, $$L_2(\cdot)$$ is linear.

Author
Contributed by Chung

=Problem 1.8 - Find the Integration Constants in Terms of the Initial Conditions=

From [[media:Pea1.f11.mtg6.djvu|Mtg 6-5]]

Given
The general structure of Linear 2nd-Order ODEs with Varying Coefficients (L2-ODE-VC) is

$$\displaystyle P(x)y''+Q(x)y'+R(x)y=F(x). $$

The general solution can be written as

$$\displaystyle y(x)=C_1y_{H1}(x)+C_2y_{H2}(x)+y_P(x). $$

Find
Find the integration constants, $$\displaystyle C_1$$ and $$\displaystyle C_2$$, in terms of the following initial conditions:

$$\displaystyle y(a)=\alpha$$ and $$\displaystyle y'(b)=\beta$$.

Solution
First, we must differentiate the solution. This yields

$$\displaystyle y'(x)=C_1y'_{H1}(x)+C_2y'_{H2}(x)+y'_P(x)$$.

Next, we set up the two equations in order to solve for the two constants.

$$\displaystyle y(a)=\alpha=C_1y_{H1}(a)+C_2y_{H2}(a)+y_P(a)$$ (Eq. 1)

and

$$\displaystyle y'(a)=\beta=C_1y'_{H1}(x)+C_2y'_{H2}(a)+y'_P(a)$$. (Eq. 2)

Then, multiply Eq. 1 by $$\displaystyle y'_{H2}(a)$$ and Eq. 2 by $$\displaystyle y_{H2}(a)$$ which yields

$$\displaystyle y'_{H2}(a)\alpha=y'_{H2}(a)C_1y_{H1}(a)+y'_{H2}(a)C_2y_{H2}(a)+y'_{H2}(a)y_P(a)$$

and

$$\displaystyle y_{H2}(a)\beta=y_{H2}(a)C_1y'_{H1}(x)+y_{H2}(a)C_2y'_{H2}(a)+y_{H2}(a)y'_P(a)$$.

Solving the two equations for $$\displaystyle C_1$$ yields

$$\displaystyle C_1=\frac{y'_{H2}(a)\alpha-y_{H2}(a)\beta+y_{H2}(a)y'_P(a)-y'_{H2}(a)y_P(a)}{y'_{H2}(a)y_{H1}(a)-y_{H2}(a)y'_{H1}(a)}$$.

Similarly, multiply Eq. 1 by $$\displaystyle y'_{H1}(a)$$ and Eq. 2 by $$\displaystyle y_{H1}(a)$$ which yields

$$\displaystyle y'_{H1}(a)\alpha=y'_{H1}(a)C_1y_{H1}(a)+y'_{H1}(a)C_2y_{H2}(a)+y'_{H1}(a)y_P(a)$$

and

$$\displaystyle y_{H1}(a)\beta=y_{H1}(a)C_1y'_{H1}(x)+y_{H1}(a)C_2y'_{H2}(a)+y_{H1}(a)y'_P(a)$$.

Solving the two equations for $$\displaystyle C_2$$ yields

$$\displaystyle C_2=\frac{y'_{H1}(a)\alpha-y_{H1}(a)\beta+y_{H1}(a)y'_P(a)-y'_{H1}(a)y_P(a)}{y'_{H1}(a)y_{H2}(a)-y_{H1}(a)y'_{H2}(a)}$$.

Author
Contributed by Allen

=References=

=Contributing Members=

title: User:Egm6321.f11.team4/HW1


text:
User:Cdiffendal7527/enes100/poppit

Link to project jupiters team page.

Give instructor top 3 project choice list
The same project has been assigned to everyone in the class. The project title is Chunk Poppit.

Write problem statement
The goal of the project is to find a way to play Poppit and eliminate all of the balloons on the screen every time. I think the best way to pop all of the balloons is to group the highest or lowest amount of colors together and pop them to see which way is more effective.

Assign Task1
I was assigned to find if there was a pattern in the randomization in which the balloons are laid out. I need to find if the there are certain amounts of colors and how many of each color balloon are grouped together.

Compare actual work done to Task 1
I played poppit fifty times using five different strategies and played the game ten times for each strategy. The averages for each game were from 17-25 balloons left in each game. In each of those games I checked to see if there were patterns in the new screen, and it is truly random. The strategies I used were popping the largest groups of one color at a time, that proved to average 23 balloons left over. i tried popping the largest groups starting from the bottom and that averaged 24 balloons left over. Then I tried to random groups of balloons containing three or more of one color, averaging 25 balloons left over. I then focused on all groups of balloons above the presents on the screen which gave me an average of 17 balloons. There was no strategy that I discovered that let me pop all of the balloons every time.

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

title: User:Cdiffendal7527/enes100/poppit


text:
User:Egm6321.f11.team4/HW2

=Problem 2.1-Verify the Given Equation=

Given
The Legendre differential equation:

When $$\displaystyle n=1 $$, it becomes:

The two homogeneous solutions for equation (1.2) are:

Find
Verify that when $$\displaystyle n=1 $$,

is true.

Solution
Substituting $$\displaystyle y $$ for $$\displaystyle x $$ in equation (1.2):

Substituting $$\displaystyle y $$ for $$\displaystyle \frac{x}{2}ln(\frac{1+x}{1-x}) $$ in equation (1.2):

Derive $$\displaystyle y^2_H = \frac{x}{2}ln(\frac{1+x}{1-x}) $$ to get its first and second order derivitives:

Substituting in equation (1.7):

So equation (1.5)is true.

Author
Contributed by Yu Chen

=Problem 2.2 - Verify the Solution of an Equation=

Given
A differential equation:

Find
is the solution to equation (2.1).

Solution
Substituting $$\displaystyle p' $$ for (2.3) in equation (2.1):

End of proof.

Author
Contributed by Yu Chen

= Problem 2.3 = From [[media:Pea1.f11.mtg7.djvu|Mtg 7-6]]

Given
The Eq. 2 in the class note [[media:Pea1.f11.mtg7.djvu|7-6]]. The Eq. 1 in the class note [[media:Pea1.f11.mtg7.djvu|7-6]].

Find
Show that the Eq. (3.1) is linear in $$\displaystyle y^{\prime}$$, and that the Eq. (3.1) is in general an N1-ODE. But the Eq. (3-1) is not the most general N1-ODE as represented by the Eq. (3.2). Give an example of a more general N1-ODE.

Show that the Eq. (3.1) is linear in the first derivative of y.
Divide the Eq. (3.1) by $$\displaystyle N(x,y) $$.

Then, the following is true. - The Eq. (3.5) proves that the Eq. (3.1) is linear in $$\displaystyle y^{\prime} $$.

Show that the Eq. (3.1) is N1-ODE.
- The Eq. (3.1) is the 1st order differential equation since the highest order of derivative is 1. - The Eq. (3.1) is Ordinary Differential equation because the differential equation includes one dependent variable and its derivatives.

To see whether the Eq. (3.1) is the linear differential equation, the superposition requirement on the Eq. (3.4) is checked. In other words, the Eq. (3.5) has to be true for the equation to be linear.

Since the RHS of the Eq. (3.6) is not equal to the RHS of the Eq. (3.7), the Eq. (3.1) does not satisfy the superposition condition.

- Since the Eq. (3.1) does not satisfy the superposition, the Eq. (3.1) is nonlinear differential equation.

The Eq. (3.1) is not the most general N1-ODE.
The Eq. (3.1) becomes the 1st order linear ordinary differential equation in particular case when the functions $$\displaystyle M(x,y)$$, $$\displaystyle N(x,y)$$ are in the form of the following.

Since the RHS of the Eq. (3.10) is equal to the RHS of the Eq. (3.11), the Eq. (3.1) satisfies the superposition condition in this particular case. - Therefore, the Eq. (3.1) is not the most general 1st order nonlinear differential equation.

Give an example of a more general N1-ODE.
An example of a general the 1st order nonlinear ordinary differential equation which cannot be converted into the particular form in the Eq. (3.1) is the following. where The Eq. (3.13) can be expressed more generally as following.

where $$\displaystyle F \left( y^{\prime} \right) $$ is any function of $$\displaystyle y^{\prime} $$ (nonlinear in general). If the function $$\displaystyle F(\cdot) $$ has no explicit inverse $$\displaystyle F^{-1}(\cdot) $$, then the Eq. (3.15) cannot be converted into the Eq. (3.1).

Author
Contributed by Shin

=''' Problem 2.4 - Show that the Eq. (3) in page 8-1 is a N1-ODE. '''= From [[media:Pea1.f11.mtg8.djvu|Mtg 8-1]]

Given
The Eq. 3 in the class note [[media:Pea1.f11.mtg8.djvu|8-1]].

Find
Show the Eq. (4.1) is non-linear 1st order ODE.

Solution
- The Eq. (4.1) is 1st order equation since the highest order of derivative is 1 as shown in the Eq. (4.2). - The Eq. (4.1) is Ordinary Differential equation because the differential equation includes one dependent variable and its derivatives. To see whether the Eq. (4.1) is the linear differential equation, the superposition requirement on the Eq. (4.3) is checked. In other words, the Eq. (4.4) has to be true for the equation to be linear.

Since the RHS of the Eq. (4.5) is not equal to the RHS of the Eq. (4.6), the Eq. (4.1) does not satisfy the superposition condition. - Since the Eq. (4.1) does not satisfy the superposition, the Eq. (4.1) is nonlinear differential equation.

Author
Contributed by Shin

=''' Problem 2.5 - Explain the Eq. (4) in p.8-1 and the Eq. (1) in p.8-2 '''= From [[media:Pea1.f11.mtg8.djvu|Mtg 8-2]]

Given
The Eq. (4) in the class note [[media:Pea1.f11.mtg8.djvu|8-1]] and the Eq. (1) in the class note [[media:Pea1.f11.mtg8.djvu|8-2]].

The Eq. (1) in the class note [[media:Pea1.f11.mtg8.djvu|7-6]] and the Eq. (2) in the class note [[media:Pea1.f11.mtg8.djvu|7-6]].

Find
Explain the Eq. (5.1) satifies the the Eq. (5.3) but can be converted to the Eq. (5.4) with the definition in the Eq. (5.2).

Solution
Move the first term in the LHS of the Eq. (5.1) to the LHS. Take the cube root on both side. Move back the term in the RHS to the LHS. Then, the Eq. (5.7) can be converted into the form of the Eq. (5.4) as shown below. when

Author
Contributed by Shin

=Problem 2.6-Show Linearly Independence= From [[media:Pea1.f11.mtg9.djvu|Mtg 9-1]]

Given
The two homogeneous solutions for equation (1.2) in problem 1.

Find
Show that the two solutions a linearly independent, and plot them.

Plotting
To make sense $$\displaystyle x $$ must be defined on the interval $$\displaystyle (-1,1) $$.

Plot $$\displaystyle y_{H}^{1} $$ and $$\displaystyle y_{H}^{2} $$. Matlab Code:

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

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Proof of Linearly Independence
To prove linearly independence of $$\displaystyle y_{H}^{1} $$ and $$\displaystyle y_{H}^{2} $$ is to show that $$\displaystyle \forall \alpha \in \mathbb R, y_{H}^{1} \ne \alpha y_{H}^{2}$$.

That is $$\displaystyle \exists \hat x $$ such that $$\displaystyle y_{H}^{1}(\hat x) \ne \alpha y_{H}^{2}(\hat x) $$.

Take $$\displaystyle \alpha = 0 $$ for example.When $$\displaystyle \alpha = 0 $$, $$\displaystyle y_{H}^{1} = \alpha y_{H}^{2} $$ yields $$\displaystyle x=0 $$.

But when $$\displaystyle x=\frac{1}{2} $$, $$\displaystyle y_{H}^{1} \ne \alpha y_{H}^{2} $$.

So $$\displaystyle \forall \alpha \in \mathbb R, y_{H}^{1} \ne \alpha y_{H}^{2}$$.

Author
Contributed by Yu Chen

= Problem 2.7 - Find and Verify the N1-ODE = From [[media:Pea1.f11.mtg9.djvu|Mtg 9-2]]

Find
And show that is an N1-ODE.

Solution
According to the particular class of N1-ODEs:
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$$ M(x,y)+N(x,y){y}'=0 \displaystyle $$ (7.3) We have
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$$ M=\frac{\partial \phi }{\partial x}=2xy^{3/2}+\frac{3}{xlog10}\displaystyle $$ (7.4) and
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 *
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$$ N=\frac{\partial \phi }{\partial y}=\frac{3}{2}x^{2}y^{1/2}+\frac{2}{ylog10}\displaystyle $$ (7.5)
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 *
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Plugging into the equation (7.3) we get
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$$ G(x,y,{y}')=(2xy^{3/2}+\frac{3}{xlog10})+(\frac{3}{2}x^{2}y^{1/2}+\frac{2}{ylog10}){y}' \displaystyle $$ (7.6)
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Obviously, we have already known that So the equation is an N1-ODE.

To be specify, an exact N1-ODE must satisfy the following 2 conditions :

Condition 1
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$$ M(x,y)+N(x,y){y}'=0 \displaystyle $$ (7.7) Condition 2
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$$ M_{y}(x,y)=0=N_{x}(x,y) \displaystyle $$ (7.8)
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For condition 1: We have proved that equation can be put in the particular form.

For Condition 2:
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$$ \frac{\partial M(x,y)}{\partial y}=3xy^{1/2} \displaystyle $$ (7.9)
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$$ \frac{\partial N(x,y)}{\partial x}=3xy^{1/2} \displaystyle $$ (7.10) Therefore
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 *
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$$ \frac{\partial M(x,y)}{\partial y}=\frac{\partial N(x,y)}{\partial x}=3xy^{1/2} \displaystyle $$ (7.11)
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 *
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In conclusion, $$ G(x,y,{y}') \displaystyle $$ is an exact N1-ODE.

Author
Contributed by Kexin Ren

= Problem 2.8 - 1st Exactness Condition = From [[media:Pea1.f11.mtg9.djvu|Mtg 9-3]]

Given
An N1-ODE as following

Find
Does the N1-ODE satisfy the first exactness condition

Solution
Express the N1-ODE above more generally as $$\displaystyle \bar M(x,y) + \bar N(x,y)\ F(y') = 0 $$ in order to find whether it can be put in the particular form.

Rearrange the equation(8.1), we have We can easily figure out that $$\displaystyle F(y')$$ has no explicit inverse $$\displaystyle F^{-1}(y')$$, so it can not be put in the particular form.

Therefore, the N1-ODE(8.1) does not satisfy the first exactness condition.

Author
Contributed by Kexin Ren

= Problem 2.9 - 2nd Exactness Condition = From [[media:Pea1.f11.mtg9.djvu|Mtg 9-4]]

Given

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$$ M=\frac{\partial \phi(x,y)}{\partial x}\displaystyle $$ (9.1)
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 *
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$$ N=\frac{\partial \phi(x,y) }{\partial y}\displaystyle $$ (9.2)
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 *
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$$ \frac{\partial M(x,y)}{\partial y}=\frac{\partial N(x,y)}{\partial x} \displaystyle $$ (9.3)
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 *
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Find
Review calculus, and find the minimum degree of differentiability of the function $$\displaystyle \phi(x,y) $$. State the full theorem and provide a proof.

Solution
Before we solute the problem, we remind the following theorems first

Mixed Derivative Theorem
If $$\displaystyle f(x,y)$$ and its partial derivatives $$\displaystyle f_{x}, f_{y}, f_{xy}$$ and $$\displaystyle f_{yx}$$ are defined in a neighborhood of $$\displaystyle (a,b)$$ and all are continuous at $$\displaystyle (a,b)$$, then $$\displaystyle f_{xy}(a,b)=f_{yx}(a,b)$$.

Mean Value Theorem
Suppose $$\displaystyle f:\mathbb {R}^2\rightarrow \mathbb {R}$$ is differentiable. Let $$\displaystyle X_{0} = (x_{0}, y_{0})$$ and $$\displaystyle X = (x_{0} + h, y_{0} + k)$$.

Then there exists $$\displaystyle C$$ which lies on the line joining $$\displaystyle X_{0}$$ and $$\displaystyle X$$ such that
 * {| style="width:100%" border="0"

$$f(X) = f(X_{0}) + f'(C)(X-X_{0}) \displaystyle $$ i.e, there exists $$ c\in (0,1) \displaystyle $$ such that
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$$f(x_{0} + h, y_{0} + k) = f(x_{0}, y_{0}) + hf_{x}(C) + kf_{y}(C) \displaystyle $$ where $$\displaystyle C = (x_{0} + ch, y_{0} + ck)$$
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 *
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Proof
Suppose $$\displaystyle f:\mathbb [a,b]\rightarrow \mathbb R$$
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$$F_{1} = f(a+h,b+k) - f(a+h,b)-f(a,b+k)+f(a,b) \displaystyle $$ (9.4) and
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$$F_{2} = f(x,y+k) - f(x,y) \displaystyle $$ (9.5) According to MVT, we can know
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$$\displaystyle \frac{F_{2}(a+h,b)-F_{2}(a,b)}{h}=\frac{\partial F_{2}}{\partial x}(a+ch,b) $$, where $$ c\in (0,1) \displaystyle $$ (9.6) Because
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$$\displaystyle {F_{2}(a+h,b)-F_{2}(a,b)}=f(a+h,b+k) - f(a+h,b)-f(a,b+k)+f(a,b)=F_{1} \displaystyle $$ (9.7) We can simply put equation(9.6) into that
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$$\displaystyle \frac{F_{1}}{h}=\frac{\partial F_{2}}{\partial x}(a+ch,b) $$ (9.8) Then focus on the right side of equation(9.8), we can get
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$$\displaystyle \frac{\partial F_{2}}{\partial x}(a+ch,b)=\frac{\partial f}{\partial x}(a+ch,b+k)-\frac{\partial f}{\partial x}(a+ch,b) $$ (9.9) According to MVT again, we have
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 *

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$$\displaystyle \frac{\frac{\partial f}{\partial x}(a+ch,b+k)-\frac{\partial f}{\partial x}(a+ch,b)}{k}=\frac{\partial^{2}}{\partial x\partial y}f(a+ch,b+ck) $$ (9.10) Back to equation(9.8),we have
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$$\displaystyle \frac{F_{1}}{hk}=\frac{\partial^{2}}{\partial x\partial y}f(a+ch,b+ck) $$ (9.11) Thus, when $$\displaystyle h\rightarrow 0$$, $$\displaystyle k\rightarrow 0$$, the limit
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$$\displaystyle \lim_{h,k\rightarrow 0}\frac{F_{1}}{hk}=\frac{\partial^{2}f}{\partial x\partial y}(a,b) $$ (9.12) The process will be totally same when we exchange the derivative order to finally get
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$$\displaystyle \lim_{h,k\rightarrow 0}\frac{F_{1}}{kh}=\frac{\partial^{2}f}{\partial y \partial x}(a,b) $$ (9.13) Since $$\displaystyle [a,b]\rightarrow \mathbb {R}$$, we have
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$$\displaystyle \frac{\partial^{2}f}{\partial x \partial y}(x,y)=\frac{\partial^{2}f}{\partial y \partial x}(x,y) $$
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 *

 * }

Author
Contributed by Kexin Ren

= Problem 2.10 - Verify thath (1) p.10-4 is indeed the solution for the N1-ODE (2) p.8-6 = From [[media:Pea1.f11.mtg10.djvu|Mtg 10-5]]

Find
Eq. (10.1) is the indeed solution for the N1-ODE (10.2)

Solution
Input Eq. (10.3) to (10.2)

As a result, (10.1) is the indeed solution for (10.2)

Author
Contributed by Chung

= Problem 2.11 - Explain why solving intergrating factor h(x,y) is usually not easy = From [[media:Pea1.f11.mtg11.djvu|Mtg 11-1]]

Find
Explain why solving (11.1) for the integrating factor h(x,y) is usually not easy.

Solution
To solve (11.1), you have to solve partial equation for $$ h_x, h_y$$ and $$M_y,  N_x$$.

Then solve intergral for exponential is not easy to calculate.

So, it is not easy to solve.

Author
Contributed by Chung

= Problem 2.12 - Find h = From [[media:Pea1.f11.mtg11.djvu|Mtg 11-3]]

Find
Find h usin (12.1)

Author
 Contributed by Chung

= Problem 2.13 - Show (3) p.11-5 = From [[media:Pea1.f11.mtg11.djvu|Mtg 11-5]]

Find
Show that

Solution
for

if c=1; h=x

Input (13.5) into (13.1)

As a result, (13.5) is a solution.

Author
 Contributed by Chung

= Problem 2.14 - Solve the general L1-ODE-VC =

From [[media:Pea1.f11.mtg6.djvu|Mtg 11-5]]

From [[media:Pea1.f11.mtg6.djvu|Mtg 12-1]]

Given
The general structure of Linear 1st-Order ODEs with Varying Coefficients (L1-ODE-VC) is

$$\displaystyle a_1(x)y'+a_0(x)y=b(x). $$

Find
The solution for $$\displaystyle y $$ if:

$$\displaystyle 1) a_1(x)=1, a_0(x)=x, b(x)=2x+3$$.

$$\displaystyle 2) $$ In general for $$\displaystyle a_1(x), a_0(x), b(x)$$.

$$\displaystyle 3) a_1(x)=x^2+1, a_0(x)=x, b(x)=2x$$.

Solution
$$\displaystyle 1) $$ If $$\displaystyle a_1(x)=1, a_0(x)=x, b(x)=2x+3$$ then $$\displaystyle y'+xy=2x+3$$ then

The integrating factor is $$\displaystyle h= \exp \int^{x}{a_0(x)ds}$$ which is $$\displaystyle h= \exp \int^{x}{sds}$$.

So, the integrating factor is $$\displaystyle h=exp\frac{x^2}{2}$$.

Using the integrating factor and eq. (1) 11-5 yields

$$ y=\exp\frac{-x^2}{2}\int^{x}{\exp\frac{-x^2}{2}*(2s+3)}ds$$.

Solving for $$y$$ yields

$$ y=\exp\frac{-x^2}{2}[2\int^{x}\exp\frac{s^2}{2}sds+3\int^{x}\exp\frac{s^2}{2}ds]$$

$$ y=2x+\exp\frac{-x^2}{2}[3\int^{x}\exp\frac{s^2}{2}ds]+C$$

$$\displaystyle 2) $$ In general for $$\displaystyle a_1(x), a_0(x), b(x)$$.

The integrating factor is $$\displaystyle h=\exp[\int^{x}\frac{a_0(s)}{a_1(s)}ds]$$.

Using eq. (1) 11-5 and replacing the integrating factor and right side function with the general terms, yields

$$\displaystyle y(x)=\frac{1}{\exp[\int^{x}{\frac{a_0(s)}{a_1(s)}}ds]}*\int^{x}\exp[\int^{x}{\frac{a_0(s)}{a_1(s)}ds}]*\frac{b(s)}{a_1(s)}ds$$.

$$\displaystyle3)$$ If $$\displaystyle a_1(x)=x^2+1, a_0(x)=x, b(x)=2x$$ then

The integrating factor is

$$\displaystyle h=\exp[\int^{x}\frac{s}{s^2+1}ds]$$.

This simplifies to

$$\displaystyle h=\exp*\frac{1}{2}\ln(x^2+1)=(x^2+1)^{1/2}$$.

So, the L1-ODE-VC becomes

$$\displaystyle (x^2+1)^{1/2}y'+(x^2+1)^{1/2}*\frac{x}{x^2+1}y=(x^2+1)^{1/2}*\frac{2x}{x^2+1}$$.

Simplifying reduces it to

$$\displaystyle (x^2+1)^{1/2}\frac{dy}{dx}+\frac{x}{(x^2+1)^\frac{1}{2}}y=\frac{2x}{(x^2+1)^\frac{1}{2}}$$.

Rearranging yields

$$\displaystyle (x^2+1)^{1/2}dy=\frac{x(2-y)}{(x^2+1)^\frac{1}{2}}dx$$.

Differenting both sides yields

$$\displaystyle (x^2+1)^\frac{1}{2}y=(2-y)*(x^2+1)^\frac{1}{2}$$.

Multiply through by $$\displaystyle (x^2+1)^\frac{1}{2}$$ and rearrange yields

$$\displaystyle y=\frac{C}{(x^2+1)^\frac{1}{2}}$$.

Author
Contributed by Allen

= Problem 2.15 - Show That K1 In (3) p.11-4 Is NOT Necessary = From [[media:Pea1.f11.mtg12.djvu|Mtg 12-2]]

Given
Eq. (3) p.11-3 is

$$\displaystyle y'+a_0(x)y=b(x)$$.

Eq. (3) p.11-4 is

$$\displaystyle h(x)=\exp[\int^x{a_0(s)ds+k_1}]$$.

Find
Show that the integration constant $$\displaystyle k_1$$ in (3) p.11-4 is NOT necessary.

Solution
Multiply (3) p.11-3 by the integrating factor, (3) p.11-4, which yields the following:

$$\displaystyle \exp[\int^x{a_0(s)ds+k_1}]*y'+\exp[\int^x{a_0(s)ds+k_1}]*a_0(x)y=\exp[\int^x{a_0(s)ds+k_1}]*b(x)$$.

Next, seperate $$\displaystyle \exp[\int^x{a_0(s)ds+k_1}]$$ into $$\displaystyle \exp[\int^x{a_0(s)ds}]*\exp[k_1]$$, which yields the following:

$$\displaystyle \exp[k_1]*\exp[\int^x{a_0(s)ds}]*y'+\exp[k_1]*\exp[\int^x{a_0(s)ds}]*a_0(x)y=\exp[k_1]*\exp[\int^x{a_0(s)ds}]*b(x)$$.

$$\displaystyle \exp[k_1]$$ cancels out of the equation showing $$\displaystyle k_1$$ is NOT necessary.

Author
Contributed by Allen

= Problem 2.16 - Show Solution from the Lecture Agrees with King 2003 = From [[media:Pea1.f11.mtg12.djvu|Mtg 12-2]]

Given
Eq. (3) p.11-3 is

$$\displaystyle y'+a_0(x)y=b(x)$$.

Eq. (3) p.11-4 is

$$\displaystyle h(x)=\exp[\int^x{a_0(s)ds+k_1}]$$.

Eq. (1) p.11-5 is

$$\displaystyle y(x)=\frac{1}{h(x)}[\int^x[h(s)b(s)ds+k_2]$$.

Find
Show that the solution of (3) p.11-3 in (1) p.11-5 agrees with the result presented in King 2003 p.512.

$$\displaystyle y(x)=Ay_H(x)+y_P(x)$$

Solution
King defines the L1-ODE-VC as

$$\displaystyle y'+P(x)y=Q(x)$$.

King also defines the integrating factor as

$$\displaystyle h(x)=\exp[\int^x{P(t)dt}]$$ where t is a dummy variable.

The equation $$\displaystyle y(x)=Ay_H(x)+y_P(x)$$ is defined with

$$\displaystyle y_H(x)=A*\exp[-\int^x{P(t)}dt]$$

$$\displaystyle y_P(x)=\exp[-\int^x{P(t)dt}]\int^x{Q(s)}\exp[\int^x{P(t)dt}]ds$$

and $$\displaystyle A$$ is constant of integration.

Expand Equation (1) P.11-5 with Equations (3) p.11-3 and (3) p.11-4 which yields

$$\displaystyle y(x)=\frac{1}{\exp[\int^x{a_0(s)ds}]}*[\int^x{[\exp[\int^x{a_0(s)ds}]*b(s)ds]+k_2}]$$

Further expand the above equation and you can break up $$\displaystyle y_H(x)$$ as

$$\displaystyle y_H(x)=\frac{k_2}{\exp[\int^x{a_0(s)ds}]}$$.

This shows that $$\displaystyle k_2=A$$ and $$\displaystyle a_0(s)=P(t)$$.

Also, it can be shown that $$\displaystyle b(s)=Q(t)$$ from the equation below:

$$\displaystyle y_P(x)=\frac{1}{\exp[\int^x{a_0(s)ds}]}*[\int^x{[\exp[\int^x{a_0(s)ds}]*b(s)ds]}]$$.

Author
Contributed by Allen

=Problem 2.17 - Homogenous and Particular solutions of general Non-Homogenous L1-ODE-VC= From [[media:Pea1.f11.mtg12.djvu|Mtg 12-2]]

Given
A General non-homogenous L1-ODE-VC (Linear, 1st order, Ordinary Differential Equation with Variable Coefficient),

$$\displaystyle P(x)\,y' + Q(x)\,y = \underbrace{R(x)}_{\color{blue}{\neq 0}}$$
 (17.1)

which can be written in the form given below,

$$\underbrace{\color{blue}{1}}_{\color{blue}{a_1(x)}} \cdot \underbrace{y'}_{\color{blue}{y^{(1)}}} + \underbrace{\frac{Q(x)}{P(x)}}_{\color{blue}{a_0(x)}}y^{\color{blue}{(0)}} = \underbrace{\frac{R(x)}{P(x)}}_{\color{blue}{b(x)}} $$
 (17.2)

The form can be written as,

$$\displaystyle y' + a_0(x)y^{(0)} = b(x) $$
 (17.3a)

Find
Find the homogenous counterpart, $$\displaystyle y_H(x) $$ of Eq. (18-3) above, by solving,

$$\displaystyle y' + a_0(x)y = 0 $$
 (17.3b)

and also find the particular solution, $$\displaystyle y_P(x) $$ by use of the method of Variation of Parameters

Solution
Eq. 17.3b can be rearranged to,

$$\displaystyle \frac{y'}{y} = - a_0(x) $$
 (17.4)

Integrate both sides, we get,

$$\displaystyle \log y = - \int^x a_0(s)ds + K_1 $$
 (17.5)

Thus homogenous solution, $$\displaystyle y_H (x) $$, can be written as,

$$\displaystyle A.y_H (x) = \exp [- \int^x a_0(s)ds ]. \underbrace{k_1}_{A} $$
 (17.6)

Thus, $$\displaystyle y_H (x) $$ is

$$\displaystyle y_H (x) = \exp [- \int^x a_0(s)ds ] $$
 (17.7)

For particular solution, $$\displaystyle y_P (x) $$ we use Variation of Parameters, by which we require y_P(x) to be following,

$$\displaystyle y_p(x) = \sum_{i=1}^{n} c_i(x) y_i(x) $$
 (17.8)

for $$ n^{th} $$ order non-linear homogenous ODE. For our case, n = 1, therefore we get,

$$\displaystyle y_p(x) = c_1(x) y_H(x) $$
 (17.9)

It is important to note for method of Variation of Parameters that above, $$\displaystyle c_1 $$ is a function of $$\displaystyle x $$, and is determined by substituting the particular solution, i.e. Eq. 17.9 into Eq. 17.3b, the ODE of interest.

Thus,

$$\displaystyle y'_H(x) + y'_P(x) + a_0(x)[y_H(x) + y_P(x)] = b(x)$$
 (17.10)

Substitute Eq. 17.9 into Eq. 17.10 to get,

$$\displaystyle y'_H(x) + [c'_1(x)y_H(x) + c_1(x)y'_H(x)] + a_0(x)y_H(x) + a_0(x)[y_H(x) + c_1(x)y_H(x)] = b(x)$$
 (17.11)

and, rearrange the terms,

$$\displaystyle \underbrace{[y'_H(x) + a_0(x)y_H(x)]}_{=0} + c'_1(x)y_H(x) + c_1(x)\underbrace{[y'_H(x) + a_0(x)y_H(x)]}_{=0} = b(x)$$
 (17.12)

Therefore $$\displaystyle c_1(x) $$ is given by,

$$\displaystyle c'_1(x) = \frac{b(x)}{y_H(x)}$$
 (17.13)

Integrate both sides w.r.t. x,

$$\displaystyle c_1(x) = \int^x \frac{b(s)}{y_H(s)} ds$$
 (17.14)

Substituting for Eq. 17.7 for expression of $$\displaystyle y_H(x) $$ into above, we will get,

$$\displaystyle c_1(x) = \int^x \underbrace{\exp [ \int^s a_0(t)dt ]}_{=h(s)} b(s) ds$$
 (17.15)

Therefore particular solution can finally be written as,

$$\displaystyle y_p(x) = [\int^x h(s) b(s) ds] \frac{1}{h(s)}$$
 (17.16)

Author
Contributed by Ankush

= Problem 2.18 - IFM for L1-ODE-VC = From [[media:Pea1.f11.mtg12.djvu|Mtg 12-5]]

Given
A particular L1-ODE-VC, (4) p.12-4

$$\displaystyle [x^4y + 10] + (\frac{1}{2} x^2)\,y'= 0 $$
 (18.1)

Find
Find if Eq. 18.1 is exact? If not, then find the integrating factor h(x) to make it exact.

Solution
Above equation, has the form as in (2), p.7-6,

$$\displaystyle \underbrace{[x^4y + 10]}_{M(x,y)} + \underbrace{(\frac{1}{2} x^2)}_{N(x,y)}\,y'= 0 $$
 (18.2)

And, above equation is linear in $$\displaystyle y' $$ and N1-ODE (Non-Linear 1st order Ordinary Differential Equation).

To check, if it is exact, we check 2nd condition for exactness, (3), p.9-3, i.e.,

$$\displaystyle M_y(x,y) = N_x(x,y) $$

Substituting the expression for $$\displaystyle M(x,y) $$ and $$\displaystyle N(x,y) $$ as shown in Eq. 18.2 into above equation,

$$\displaystyle \frac{\partial}{\partial x} (x^4y + 10)  =? \ne \frac{\partial}{\partial x} (\frac{x^2}{2}) $$
 (18.3)

Where, above $$\displaystyle =?\ne $$ denotes that it can be either $$\displaystyle = $$ or $$\displaystyle \ne $$, Upon further differentiation we get,

$$\displaystyle x^4 \ne x \forall x $$

Therefore the given equation is not exact, and we use Euler Integrating Factor Method (IFM) to find h, integrating factor to make the given equation exact. For this we notice that,

$$\displaystyle - \frac{1}{N}(N_x - M_y) = - \frac{1}{\frac{x^2}{2}}(x-x^4) = (-\frac{2}{x} + 2x^2) \rightarrow n(x) $$
 (18.4)

Above given expression is thus a function of $$\displaystyle x $$ only, (2) p.11-2 and this means that h is a function of x only. Therefore we can integrate,

$$\displaystyle \frac{h_x}{h} = (- \frac{2}{x} + 2x^2) dx $$
 (18.5)

to get,

$$\displaystyle \int^x \frac{h_x}{h} = \int^x (- \frac{2}{x} + 2x^2) dx $$
 (18.6)

From above equation, we can solve for $$\displaystyle h $$,

$$\displaystyle h(x) = exp[\int^x (- \frac{2}{x} + 2x^2) dx + k] $$
 (18.7)

where, k is integration constant. Above expression can be simplified in following steps to get the final expression of integration factor, h which will make the given L1-ODE-VC Eq. 18.1 exact,

$$\displaystyle h(x) = exp[-2\log(x) + \frac{2}{3}x^3] $$

$$\displaystyle h(x) = exp(-2\log(x)).exp(\frac{2}{3}x^3) $$

And, thus the final form of $$\displaystyle h(x) $$, the integrating factor is,

$$\displaystyle h(x) = exp(\frac{2}{3}x^3).\frac{1}{x^2} $$
 (18.8)

By multiplying both sides of Eq. 18.1 by $$\displaystyle h(x) $$, it can be verified that the second condition of exactness, (3)p.9-3 is now satisfied.

Author
Contributed by Ankush

= References =

=Contributing Members=

title: User:Egm6321.f11.team4/HW2


text:
User:Ayapici/enes100/poppit

User:Ayapici/enes100/poppit Link to project jupiters team page.

Project Choice
The project choice was not made by students this time, everybody was assigned to do the same project and it is called "Chunk Poppit."

Write problem statement
The major question about this project is that at first how can we play the game poppit and clear all the balloons on the screen? my hypothesis is to find the all the statistics about the game. My goals is to create something simple and possible to beat the game.

Assign Task1
My task was at first to play the game and understand how it works and right down some notes and collect some data as in like the amount of balloons, columns, rows and the colors.I have collected all of the basic data so far, there are 150 balloons in total, and there are 15 columns, 10 rows and 5 different colors ( red,green, blue, yellow, purple). After that I have played the game with trying different strategies to eliminate more balloons and have as few as possible.

Compare actual work done to Task1
I have compared my work and my strategies with my team members and shared my opinion. They have come up with pretty much similar thoughts. The task one was not difficult, but it is was steps that must have been done to go further.

adding photos
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

title: User:Ayapici/enes100/poppit


text:
User:Egm6321.f11.team4.YuChen

=Report Assignments= Report 1 Report 2

title: User:Egm6321.f11.team4.YuChen


text:
User:Egm6321.f11.team4.YuChen/HW1

= Problem 1.1 - First and Second Derivative of the Equation of Maglev Trains =

Given
The equation of the meglev trains:$$\displaystyle f(Y^1 (t),t) $$

Find
Derive the first and second time derivatives for the given equation.

Derivation of Multivariable Functions
Suppose $$\displaystyle u(x) $$ and $$\displaystyle v(x) $$ are functions of $$\displaystyle x $$. Then the first partial derivative with respect to $$\displaystyle x $$ of the multivariable function $$\displaystyle f(u,v)=f(u(x),v(x)) $$ is:

First Derivative
Now let $$\displaystyle s=Y^1(t) $$ for ease,so the first time derivative of the given function should be:

Use $$\displaystyle \dot Y^1(t) = \frac{dY^1}{dt} $$ for ease:

Second Derivative
Derive $$\displaystyle (Eq. 1) $$ with respect to $$\displaystyle t $$ again:

For ease, use $$\displaystyle f_{,x} := \frac{\partial f}{\partial x} $$ and $$\displaystyle f_{,xy} := \frac{\partial^2 f}{\partial x \partial y} $$, so (1.3) is actually:

=Problem 2.1-Show the Relationship of the Derivatives in Problem 1 and the Reynolds Transport Theorem & the Derivation of Coriolis Force=

Given
$$\displaystyle (Eq.1) $$ and $$\displaystyle (Eq.2) $$ in problem 1.

Find
Show the relationship of $$\displaystyle (Eq.1) $$ and the Reynolds Transport Theorem in Continuum Mechanics. Show the similarity of $$\displaystyle (Eq.2) $$ and the Derivation of Coriolis Force.

Reynolds Transport Theorem
Consider the function $$\displaystyle f=f(x(t),t) $$ over the time-dependent region $$\displaystyle \Omega(t) $$ that has boundary $$\displaystyle \partial \Omega(t) $$, then take the derivative with respect to time $$\displaystyle t $$:

Now we wish to move the derivative under the integral sign.

In continuum mechanics this theorem is often used for material elements, which are parcels of fluids of solids which no material enters or leaves. Now suppose $$\displaystyle \Omega(t) $$ is a 3-dimensional region (higher dimension also applies). $$\displaystyle \mathbf{v} $$ is the velocity field of such material element, $$\displaystyle \Sigma $$ is the

NOT FINISHED

=Problem 1.6 - Proof of Nonlinearity of Certain Function= From [[media:Nm1.s11.mtg23.djvu|Mtg 23-1]], [[media:Nm1.s11.mtg26.djvu|Mtg 26-1]]

Given
The first term in EOM:

where

Find
Show function (6.1) is nonlinear with respect to $$\displaystyle Y^1 $$.

Definition of Nonlinearity
If a function $$\displaystyle F(x) $$ does notsatisfy:

then it is nonlinear.

Proof of Nonlinerarity
Let $$\displaystyle F(x)=M[1-\bar{R} u_{,ss}^2 (x,t)] \ddot x $$, then:

If $$\displaystyle F(\alpha x + \beta y) = \alpha F(x) +\beta F(y) $$ is true, then:

is true for any real number $$\displaystyle \alpha, \beta $$ and any $$\displaystyle x, y $$ in the domain. That means:

is true for any real number $$\displaystyle \alpha, \beta $$ and any $$\displaystyle x, y $$ in the domain, which is absolutely not true. So equation (6.3) is not true for $$\displaystyle c_3(Y^1,t) \ddot Y^1 $$, and $$\displaystyle c_3(Y^1,t) \ddot Y^1 $$ is nonlinear.

title: User:Egm6321.f11.team4.YuChen/HW1


text:
User talk:Hkang0839

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title: User talk:Hkang0839


text:
User:BobFK/enes100

Link back to my current Project page.

Individual documentation for all enes100 introduction to engineering projects fall 2011.

title: User:BobFK/enes100


text:
User:BobFK/enes100/poppit

Link to Team page: 550_prgm

Give instructor top 3 project choice list
1) Engines

2) Poppit

3) Ball Bot

Problem Statement
Is the software 'Processing' an adequate platform to optimize automated solution of the game Poppit?

Task One
Explore the qualities Poppit requires of Processing.

Kinks
Firefox does not support Poppit on-campus. I developed and attempted several remedies, however it was my teammate who reminded me that Poppit can be run from Google Chrome.

Narrative
There are 150 initial balloons, of which around 30 are, on average, a given color. Also by average, 30 legal starting moves exist. Simple statistical knowledge tells me that this should represent approximately 30! legal moves, or 2.65*10^32 in scientific notation.

The order of possible moves indicates that Poppit cannot be easily solved with brute force. My next inclination therefore is to study estranged groupings of balloons. Focusing on a given side of the game, I can consistently eliminate one or more columns by observing which balloons are estranged from their neighbors. This leads me to believe that a positional relation between like-colored balloons can eventually reduce Poppit to a more manageable problem. I have also noted that for every game in which each balloon is paired, there exists a solution for which no balloons remain. This does Not, as it may seem to say, imply that there is always a way to pop every balloon. Merely the set of balloon arrangements for which every balloon is paired has such a solution. This principle can be applied to a game as a whole or to a subset, thereby directly reducing computational complexity. Processing's capability produce to these calculations should be sound. I would be capable of working them in Java, the language on which Processing is based.

Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Task Two
I've been assigned to Engines/Howard Community College/Fall2011/550_ARM and will postpone this project for the duration.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:BobFK/enes100/poppit


text:
User:EGM6321.f11.team1.pan

Homework Assignments
EGM6321.f11.team1.pan/HW1

EGM6321.f11.team1.pan/HW2

EGM6321.f11.team1.pan/HW3

EGM6321.f11.team1.pan/HW4

EGM6321.f11.team1.pan/HW5

EGM6321.f11.team1.pan/HW6

EGM6321.f11.team1.pan/HW7

Team Report
Team1 report

Lecture Video
Lecture video

Sample Table
click here

title: User:EGM6321.f11.team1.pan


text:
User talk:EGM6321.f11.team1.pan

 Hello EGM6321.f11.team1.pan, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:EGM6321.f11.team1.pan


text:
User talk:Egm6321.f11.team4.YuChen

 Hello Egm6321.f11.team4.YuChen, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Egm6321.f11.team4.YuChen


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User talk:Andyn6361

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title: User talk:Andyn6361


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User talk:Ayapici

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User talk:Cdiffendal7527

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title: User talk:Cdiffendal7527


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Category:Conservation

This category is for pages related to conservation.

title: Category:Conservation


text:
User:Egm6321.f11.team3/Hwk1

Given
From lecture note 2-4, first total time derivative is given by,
 * $$\displaystyle

\frac{\mathrm{d} }{\mathrm{d} t}f(Y^{1}(t),t)=\frac{\partial f(Y^{1}(t),t)}{\partial S}\dot{Y^{1}}+\frac{\partial f(Y^{1}(t),t)}{\partial t} $$ with
 * $$\displaystyle

\dot{Y^{1}}:=\frac{\mathrm{d} Y^{1}(t)}{\mathrm{d} x} $$

Find
Show that
 * $$\displaystyle

\frac{\mathrm{d^2}f }{\mathrm{d} t^2}=f_{,S}(Y^{1},t)\ddot{Y}^1+f_{,SS}(Y^{1},t)(\dot{Y}^{1})^{2}+2f_{,St}(Y^{1},t)\dot{Y}^{1}+f_{,tt}(Y^{1},t) $$ with
 * $$\displaystyle

f_{,S}(Y^{1},t):=\frac{\partial f(Y^{1},t)}{\partial S} $$
 * $$\displaystyle

f_{,St}(Y^{1},t):=\frac{\partial^{2} f(Y^{1},t)}{\partial S \partial t} $$ Similarly for the other partial derivatives

Solution
We solved this problem on our own Let
 * $$\displaystyle

\frac{\mathrm d^{2}f}{\mathrm dt^{2}}=\frac{\mathrm d}{\mathrm dt}(\frac{\mathrm d}{\mathrm dt}f(Y^{1}(t),t))=\frac{\mathrm d}{\mathrm dt}(\frac{\partial f(Y^{1}(t),t)}{\partial s}\dot{Y}^{1}+\frac{\partial f(Y^{1}(t),t)}{\partial t}) $$
 * $$\displaystyle

=\frac{\partial f(Y^{1}(t),t)}{\partial S}\frac{\mathrm d^{2}Y^{1}(t)}{\mathrm dt^{2}}+\frac{\partial }{\partial s}(\frac{\partial f(Y^{1}(t),t)}{\partial s}\frac{\mathrm dY^{1}(t)}{\mathrm dt}+\frac{\partial f(Y^{1}(t),t)}{\partial t})\frac {\mathrm dY^{1}(t)}{\mathrm dt} $$
 * $$\displaystyle

+\frac{\partial}{\partial t}(\frac{\partial f(Y^{1}(t),t)}{\partial S}\frac{\mathrm dY^{1}(t)}{\mathrm dt}+\frac{\partial f(Y^{1}(t),t)}{\partial t}) $$
 * $$\displaystyle

=\frac{\partial^{2}f(Y^1(t),t)}{\partial S^{2}}(\frac{\mathrm dY^{1}(t)}{\mathrm dt})^{2}+2\frac{\partial^{2}f(Y^{1}(t),t)}{\partial S \partial t}\frac{\mathrm dY^{1}(t)}{\mathrm dt}+\frac{\partial^{2}f(Y^{1}(t),t)}{\partial t^{2}} $$
 * $$\displaystyle

+\frac{\mathrm d^{2}Y^{1}(t)}{\mathrm dt^{2}}\frac{\partial f(Y^{1}(t),t)}{\partial S} $$
 * $$\displaystyle

=f_{,S}(Y^{1},t) \ddot{Y}^{1}+f_{,SS}(Y^{1},t)(\dot{Y}^{1})^{2}+2f_{,St}(Y^{1},t)\dot{Y}^{1}+f_{,tt}(Y^{1},t) $$

Given
The equation of motion (EOM) for the maglev train is modeled by:

 * $$\displaystyle

\displaystyle f(S,t)| _{S=Y^1(t)}=f(Y^1(t),t) $$ From lecture notes on page 2-4, the 1st total time derivative is given by:

and the 2nd total time derivative, on page 2-5, is given by:

Find

 * Derive the 1st total time derivative.
 * Derive the 2nd total time derivative.
 * Show the similarity with the derivation of the Coriolis force.

Solution, 1st Total Time Derivative
Applying the chain rule to
 * $$\displaystyle

f(S,t) $$

Yields:
 * $$\displaystyle

\frac{df(S,t)}{dt}=\frac{\partial f}{\partial S}\frac{\partial S}{\partial t}+\frac{\partial f}{\partial t}\frac{\partial t}{\partial t} = \frac{\partial f(S,t)}{\partial S}\frac{\partial S}{\partial t} + \frac{\partial f(S,t)}{\partial t}\frac{\partial t}{\partial t} $$ Given that $$ \displaystyle S $$ is evaluated at $$ \displaystyle Y^1(t) $$ :
 * $$\displaystyle

f(S,t)| _{S=Y^1(t)}=f(Y^1(t),t) $$ Then the equation can be rewritten as:

And simplified to result in equation 2.1:
 * $$\displaystyle

\frac{d}{dt}f(Y^1(t),t)= \frac{\partial f(Y^1(t),t)}{\partial S} \dot Y^1+\frac{\partial f(Y^1(t),t)}{\partial t} $$

Solution, 2nd Total Time Derivative
The general expression for the 2nd total time derivative is:

 * $$\displaystyle

\frac{d^2}{dt^2}f=\frac{d}{dt}\left(\frac{d}{dt}f\right) $$ Substituting from equation 2.3 results in:
 * $$\displaystyle

\frac{d^2}{dt^2}f(Y^1(t),t)=\frac{d}{dt}\left(\frac{\partial f}{\partial S}\frac{\partial S}{\partial t} + \frac{\partial f}{\partial t}\underbrace{\frac{\partial t}{\partial t}}_{1}\right) $$ Applying the chain rule yields:
 * $$\displaystyle

\frac{d^2}{dt^2}f(Y^1(t),t)=\frac{\partial^2f}{\partial S^2}\left(\frac{\partial S}{\partial t}\right)^2

+\left(\frac{\partial^2 f}{\partial S \partial t} \cdot \frac{\partial S}{\partial t} \cdot \underbrace{\frac{\partial t}{\partial t}}_{1}\right)

+\left( \frac{\partial f}{\partial S} \cdot \underbrace{\frac{\partial^2 S}{\partial S \partial t}}_{0} \cdot \frac{\partial S}{\partial t}\right)

+\left(\frac{\partial f}{\partial S} \cdot \frac{\partial^2 S}{\partial t^2} \cdot \underbrace{\frac{\partial t}{\partial t}}_{1}\right)

+\left(\frac{\partial^2 f}{\partial S \partial t} \cdot \frac{\partial S}{\partial t}\right)

+\left(\frac{\partial^2 f}{\partial t^2} \cdot \underbrace{\frac{\partial t}{\partial t}}_{1}\right)

$$ Collecting terms and simplifying creates the equation:
 * $$\displaystyle

\frac{d^2}{dt^2}f(Y^1(t),t)=\underbrace{\frac{\partial^2f}{\partial S^2}}_{f_{,SS}}\left(\underbrace{\frac{\partial S}{\partial t}}_{\dot Y^1}\right)^2

+\left(2 \cdot \underbrace{\frac{\partial^2 f}{\partial S \partial t}}_{f_{,St}} \cdot \underbrace{\frac{\partial S}{\partial t}}_{\dot Y^1}\right)

+ \left( \underbrace{\frac{\partial f}{\partial S}}_{f_{,S}} \cdot \underbrace{\frac{\partial^2 S}{\partial t^2}}_{\ddot Y^1} \right)

+ \underbrace{\frac{\partial^2 f}{\partial t^2}}_{f_{,tt}}

$$ Then substituting using the denoted subscripts, and rearranging, results in equation 2.2:
 * $$\displaystyle

\displaystyle \frac{d^2f}{dt^2}=f_{,S}\left({Y^1,t}\right)\ddot Y^1+f_{,SS}(Y^1,t)(\dot Y^1)^2+2f_{,St}(Y^1,t)\dot Y^1+f_{,tt}(Y^1,t) $$

Similarity to Coriolis Force Derivation
Suppose that a given object has position vector r in the rotating reference frame with instantaneous velocity v. Then it is obviously that $$\frac{\mathrm{d} r}{\mathrm{d} t}=\frac{\partial r}{\partial t}+\dot{\theta}\times r=v+\omega\times r $$ $$\frac{\mathrm{d}^{2} r}{\mathrm{d} t^{2}}=\frac{\mathrm{d}}{\mathrm{d}t}(\frac{\partial r}{\partial t}+\dot{\theta}\times r)=\frac{\partial}{\partial t}(\frac{\partial r}{\partial t}+\dot{\theta}\times r)+\frac{\mathrm d}{\mathrm d t}(\dot{\theta}\times r)$$ $$ =\frac{\partial^{2}r}{\partial t^{2}}+2\dot{\theta}\times\frac{\partial r}{\partial t}+\ddot{\theta}\times r+\dot{\theta^{2}}\times r $$ Compare 1st and 2nd Total Time Derivatives with Coriolis equation we can find that

Given
The horizontal force $$\displaystyle c_0 $$ for the wheel/magnet on the maglev train is:

Find
Analyze the dimensions of all terms in equation 3.1 and provide the physical meaning.

Solution
We solved this problem on our own The horizontal force $$\displaystyle c_0 $$ for the wheel/magnet on the maglev train is:

Dimensions of all the quantities:

[F1] = Force (F) $$[\bar R] = Length (L)$$ [u2,ss] = L-1 [F2] = F [u2,s] = 1 [T] = FL [R] = L [M] = Mass (M) [u1,tt] = Lt-2 [u2,stt] = t-2 [c0] = F A := $$\displaystyle F^1[1-\bar R u^2_{,SS} (Y^1,t)] $$ Dimension of A = [A] [A] = -F [1-LL-1] [A] = F B := F2u2,s Dimension of B = [B] [B] = F x 1 = F [B] = F C := $$\displaystyle \frac{T}{R} $$ Dimension of C = [C] [C] = FL x L-1 [C] = F

D := $$\displaystyle M \left [(1-\bar R u^2_{,SS})(u^1_{,tt}-\bar R u^2_{,Stt})+u^2_{,S} u^2_{,tt} \right]$$ Dimension of D = [D] [D] = M{( 1-LL-1)(Lt-2-Lt-2) + 1 x Lt-2 } [D] = M {1 x Lt-2 + Lt-2 } [D] = MLt-2 [D] = F  [A]=[B]=[C]=[D]=F [c0(Y1,t)] = F Physical Meaning A is the transverse force due to deformation in the transverse direction B is the transverse force due to deformation of guideway slope C is the force due to rotation of wheel D is the transverse force due to acceleration of wheel c0(Y1,t) is the sum of all forces acting on the wheel in the transverse direction. Prashant Gopichandran 00:47 September 14th,2011 (UTC)

Given
ξ=(ξ1,ξ2,ξ3)

X(ξ)=X1(ξ1)X2(ξ2)X3(ξ3)

Find
Draw the polar coordinate lines, in a 2-D plane emanating from a point, not at the origin.

Solution
We solved this problem on our own In a 2-D plane, choose a point p and call it the pole, as shown in figure below. Then we select an x axis through that pole and let x axis parallel to x1 axis. It is polar coordinate lines emanating from a point, not at the origin.

Reference
Polar Coordinates

Find
Show that (5.1) becomes:
 * $$\displaystyle

y''+\underbrace{\frac{g'(x)}{g(x)}}_{a_1(x)}y'+a_0(x)\,y=0 $$ Change symbols to simplify (lecture notes 4-5): $$\displaystyle \xi_i \rightarrow x$$ $$X_i(\xi_i)\rightarrow y(x)$$ $$g_i(\xi_i)\rightarrow g(x)$$ $$f_i(\xi_i)\rightarrow a_0(x)$$

Solution
Substitute into equation 5.1 with these new variables: $$\displaystyle \underbrace{\frac{1}{g_i(\xi_i)}}_{\color{blue}\frac{1}{g(x)}} \underbrace{\frac{d}{d \xi_i}}_{\color{blue}\frac{d}{dx}} \left[\underbrace{g_i(\xi_i)}_{\color{blue}g(x)} \underbrace{\frac{d X_i(\xi_i)}{d \xi_i}}_{\color{blue}\frac{d y(x)}{dx}} \right]+ \underbrace{f_i(\xi_i)}_{\color{blue} a_0 (x)} \underbrace{X_i(\xi_i)}_{\color{blue} y(x)}=0 $$ $$\displaystyle \frac{1}{g(x)} \cdot \frac{d}{dx} \cdot \left[g(x) \cdot \frac{dy(x)}{dx}\right]+a_0 (x) \cdot y(x)=0 $$ $$\displaystyle \frac{1}{g(x)} \cdot \left[g'(x) \cdot \frac{dy(x)}{dx} + g(x)\frac {d^2y(x)}{dx^2}\right]+a_0 (x) \cdot y(x)=0 $$ $$\displaystyle \frac{g'(x)}{g(x)} \cdot \frac{dy(x)}{dx} + \frac {d^2y(x)}{dx^2}+a_0 (x) \cdot y(x)=0 $$ $$\displaystyle \underbrace{\frac{g'(x)}{g(x)}}_{a_1(x)} \cdot \underbrace{\frac{dy(x)}{dx}}_{y'} + \underbrace{\frac {d^2y(x)}{dx^2}}_{y''}+a_0 (x) \cdot y(x)=0 $$ $$\displaystyle y''+a_1(x)y'+a_0(x)\,y=0 $$

Given:
The equation of motion of the wheel/magnet provide in Lecture 3, p3-3 is given as:

where $$ \displaystyle Y^1 $$ is the nominal position of the wheel.

The mass component of the equation of motion is defined as

Find:
Show that $$\displaystyle c_3(Y^1,t)\ddot Y^1$$ is nonlinear with respect to $$\displaystyle Y^1 $$.

Solution
We solved this problem on our own and checked it against previous class solutions A linear seoond order equation is one that can be written in the form:

or with any function y with a continuous second derivative on the interval I, then

To demonstrate $$\displaystyle c_3(Y^1,t)\ddot Y^1$$ to be non-linear with respect to $$\displaystyle Y^1 $$, it suffices to show that the equation 6.6 is not always satisfied.

Now we insert Equation 6.2

and change the notation of the second derivative

Now we pull out the constant on the left side of the equation.

It is now clear that the two sides of the equation are not equal due to the constant on the second internal term. Therefore, $$\displaystyle c_3(Y^1,t)\ddot Y^1$$ is nonlinear with respect to $$\displaystyle Y^1 $$.

Given

 * $$\displaystyle

L_{2}(\cdot )=\frac{d^{2}(\cdot )}{dx^{2}}+a_{1}(x)\frac{d(\cdot )}{dx}+a_{0}(x)(\cdot ) $$ You can see from [PEA1.F11.mtg6 p6-3 (1)]

Find
Show that :$$\displaystyle L_{2}(\cdot )$$ is linear

Solution
We solved this problem on our own Let u(x) and v(x) be any 2 functions of x Let :$$\displaystyle \alpha \, and \, \beta $$ be and 2 real numbers We write :$$\displaystyle \forall u,v:\mathbb{R}\rightarrow \mathbb{R} \, x \mapsto u(x)  \,\,  x \mapsto v(x)  \,\, \forall \alpha ,\beta \in \mathbb{R} $$

So it is linear.

Given

 * $$\displaystyle

y(x)=y^{_{H}}(x)+y^{_{P}}(x)=c_{1}y_{H}^{1}(x)+c_{2}y_{H}^{2}(x)+y_{P}(x) $$

Find
Find the integration constants in terms of the initial conditions :$$\displaystyle y(a)=\alpha \,\, \, y{}'(a)=\beta $$

Solution
We solved this problem on our own

 * $$\displaystyle You\,\, can\, get\,  c_{1}\, by\,\,  equation(8.1)\times \dot{y}_{H}^{2}(a)-equation(8.2)\times y_{H}^{2}(a)$$

 * $$\displaystyle You\,\, can\, get\,  c_{2}\, in\,\,   the\,  similar\,  way$$

 * $$\displaystyle \Rightarrow c_{2}= \frac{\alpha \dot{y}_{H}^{1}(a)-\beta y_{H}^{1}(a)-y_{P}(a)\dot{y}_{H}^{1}(a)+\dot{y}_{P}y_{H}^{1}(a)}{y_{H}^{2}(a)\dot{y}_{H}^{1}(a)-\dot{y}_{H}^{2}(a)y_{H}^{1}(a)}       $$

 (8.4)

title: User:Egm6321.f11.team3/Hwk1


text:
User talk:Kayhh~enwikiversity

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Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

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Renamed
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title: User talk:Kayhh~enwikiversity


text:
User talk:Ckpress

 Hello Ckpress, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Ckpress


text:
User talk:ARoberts

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 18:46, 7 September 2011 (UTC)

title: User talk:ARoberts


text:
User talk:Jkim7610

 Hello Jkim7610, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 18:49, 7 September 2011 (UTC)

title: User talk:Jkim7610


text:
User talk:Mderrick4950

 Hello Mderrick4950, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 18:49, 7 September 2011 (UTC)

title: User talk:Mderrick4950


text:
User talk:Baminnick

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title: User talk:Baminnick


text:
User talk:Ndelaney9295

 Hello Ndelaney9295, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Ndelaney9295


text:
User:Egm6321.f11.team4.ankushbhatia

= Homework assignments= HW1

title: User:Egm6321.f11.team4.ankushbhatia


text:
Template:Renameuser

→

 * Datestamp: ~
 * Reason:

title: Template:Renameuser


text:
Template:Renameuser2

 * For bureaucrat use: [ rename user ]
 * For bureaucrat use: [ rename user ]
 * For bureaucrat use: [ rename user ]

title: Template:Renameuser2


text:
Template:User23

User: ( talk · contribs · deleted · · global account browser ·  central auth  ·  logs  · [ target logs ] )

title: Template:User23


text:
Template:Listuser

[  ]

title: Template:Listuser


text:
User:Kjaved373/enes100

Engineering class at Howard Community College Fall 2011.

List of engineering projects:

 * Poppit
 * Electric Cars
 * Rubberband Gun
 * Rubberband Gun - Reloaded

title: User:Kjaved373/enes100


text:
User:Anton1102

 * popit
 * Gstrandbeest
 * Rubberband Gun
 * Rubberband Gun 2

title: User:Anton1102


text:
User:Kjaved373/enes100/poppit

Link to 550_prgm team page.

Write problem statement
Is the program "Processing" a suitable choice for finding a solution to the Poppit game?

Assign Task1
Learn how "Processing" works and create a simple program using examples.

Compare actual work done to Task1
A couple of examples from "Processing" were ran through. The simple program that was made was a single player pong like game using the 'bounce' example.

Week1 Narrative
Added two screenshots of 'Pong' game on project page. Edited and completed Wikiversity project page for our team. A couple problems encountered were mostly programming the 'Pong' game and editing Wikiversity pages. These were solved by doing research online and looking at tutorials and examples included.

title: User:Kjaved373/enes100/poppit


text:
User:Anton1102/enes100

anton1102 engineering projects for forester class

title: User:Anton1102/enes100


text:
User:Jhasslinger/enes100

This is going to be a list of all engineering projects I participated in at Howard Community College, fall 2011

Projects
Current Project:
 * HCJ
 * HCJ Team Page

Past Prjects:
 * Poppit
 * Poppit Team
 * HAM
 * HAM Team
 * HAM_V2
 * HAM_V2 Team

title: User:Jhasslinger/enes100