text:
User talk:TRamnarine

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title: User talk:TRamnarine


text:
User talk:Mgrollman~enwikiversity

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Your account will be renamed
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.

Unfortunately, your account clashes with another account also called Mgrollman. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Mgrollman~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:38, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:16, 19 April 2015 (UTC)

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

title: User talk:Mgrollman~enwikiversity


text:
User talk:Wissutak

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title: User talk:Wissutak


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User talk:Sandysansine

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title: User talk:Sandysansine


text:
User talk:EAS

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title: User talk:EAS


text:
User talk:Htaikseng

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title: User talk:Htaikseng


text:
User talk:Cutedatchy

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title: User talk:Cutedatchy


text:
User talk:Seedy

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title: User talk:Seedy


text:
User talk:Pemika 5080693

Pemika ID 5080693 Sec. 1

The specs for the best PC computer

INTEL Performance PC

1.	CPU : Intel i7-940 2.93 GHz

2.	Mainboard : ASUS P6T Deluxe X58+ICH10R

3.	Memory : OCZ DDR3 Gold Edition 6GB 1600MHz Triple Channel (2GBx3) 8-8-8-24

4.	Hard Disk : Barracuda ES.2 ST3500320NS 500GB SATA-II 32MB NCQ (Enterprise Drive)

5.	Graphics Card : XFX GTX285 ZDFF 1024MB DDR3 512

6.	Monitor : Samsung LCD 22" T220

7.	Optical Drive : ASUS DRW2014L1T 20X, SATA-Interface

Other equipment

1.	computer keyboard : .Optimus Maximus

2.	Power supply : 600 WATT POWER SUPPLY Intel AMD PC Desktop Computer

3.	Mouse : The Logitech Gaming Mouse G500

4.	Modem : Modem (Intel 82536 Chipset)

5.     Sound card : Creamware- SCOPE home ( LUNA II )

6.	Video card : HP ATI Radeon X1300 Video Card

7.	CD/DVD : HP/Compaq Pavilion DV1000, DV1100 & DV1300 Dual Format 8X DVD+/-RW Drive

8.	Case : Case RaidMax SAGITTA (921WB) (pro) (No Power)

9.	Floppy drive : Intel (AXXUSBFLOPPY) (AXXUSBFLOPPY) Internal Floppy Drive

Pemika 5080693 05:01, 3 October 2009 (UTC)

title: User talk:Pemika 5080693


text:
User talk:Supinya

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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! --Mu301Bot 16:45, 25 September 2009 (UTC) Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 16:45, 25 September 2009 (UTC)

the first question is how many operating sysrems?? the answer is there are about 5 operating systems which are BSD, Darwin (Mac OS X), Linux, SunOS (Solaris/OpenSolaris), and Windows NT (XP/Vista/7).

the second question is which is suitable for netbook. the answer is Windows 7.

title: User talk:Supinya


text:
User talk:Kgaipo 5790

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title: User talk:Kgaipo 5790


text:
User talk:Sunnamcg

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title: User talk:Sunnamcg


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User talk:Yourockmyworld

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title: User talk:Yourockmyworld


text:
User talk:Nemma

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title: User talk:Nemma


text:
User talk:Gsiemens

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title: User talk:Gsiemens


text:
User talk:Odelia Kadi

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title: User talk:Odelia Kadi


text:
User talk:Brianna Iowa

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title: User talk:Brianna Iowa


text:
User talk:Arieli Hanner

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title: User talk:Arieli Hanner


text:
User talk:Napoleon Astro

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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! --Mu301Bot 16:46, 25 September 2009 (UTC) Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 16:46, 25 September 2009 (UTC)

title: User talk:Napoleon Astro


text:
User talk:Wendy Ulbani

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title: User talk:Wendy Ulbani


text:
User talk:Nila Ming

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title: User talk:Nila Ming


text:
User talk:Lance Nelson

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title: User talk:Lance Nelson


text:
Digital Libraries/Evaluation and user studies

 * Older versions of the draft developed by UNC/VT Project Team (2009-10-07 PDF WORD)

Module name
Digital Library Evaluation, User Studies

Scope
While a number of kinds of evaluation/research studies may be conducted during the design and development of a digital library (e.g., usability testing), this module is concerned with methods for evaluating the outcomes, impacts, or benefits of a digital library, including cost/benefit analyses. It also includes methods that are useful for general user studies (i.e., studies that intend to more fully understand people's interactions with digital libraries). While some methods covered here are useful for usability testing, usability inspections and usability testing are explicitly covered in module 6-d, Interaction Design, Information Summarization and Visualization, and Usability Assessment.

Learning objectives:
By the end of this module, the student will be able to:
 * a. Understand the importance of DL evaluation;
 * b. List and describe the strengths and weaknesses of multiple approaches to evaluation; and
 * c. Apply an appropriate evaluation method to a particular DL.

5S characteristics of the module:

 * a. Streams: N/A
 * b. Structures: N/A
 * c. Spaces: N/A
 * d. Scenarios: Scenarios may form the basis of an evaluation plan, by describing particular situations of use that must be supported effectively by the DL.
 * e. Societies: The concept of societies may be useful in planning an evaluation because it will support the evaluator in more systematically consider the potential stakeholders of the DL.

Level of effort required:

 * a. In-class time: 2-2 1/2 hours
 * b. Out-of-class time: 1 1/2 hours for assigned reading
 * c. Learning activities (optional): See notes on timing with each activity or assignment.

Relationships with other modules:

 * a. It is expected that this module will follow other modules on digital libraries, and will be in the final portions of the module sequence.

Prerequisite knowledge required:

 * a. Students will not be expected to have had prior training in social science research methods.

Introductory remedial instruction:

 * a. None

Evaluation and user studies

 * 1. Definition of evaluation: "An appraisal of the performance or functioning of a system, or part thereof, in relation to some objective(s)" Saracevic, 2000, p.359
 * a. Evaluation incorporates the making of value judgments about whether performance is adequate
 * b. Its purpose is to inform decision making (Reeves et al., 2003)
 * c. Evaluation is critical to any project
 * d. NSF recommended that at least 10% of the project budget be devoted to evaluation in their early DL initatives

 * 2. User studies may be more general, in terms of the types of questions asked
 * a. They do not necessarily incorporate the making of value judgments about performance quality

 * 3. User studies may be more specific, in that they involve users
 * a. Evaluations may be conducted on the DL collection or other aspects of the DL without involving users

 * 4. This module will focus particularly on evaluations that involve users

The object of the evaluation or user study: digital libraries and their processes/functions

 * 1. A particular aspect of a digital library
 * 2. Individual digital libraries
 * 3. Multiple digital libraries

 * 4. DL processes that may be evaluated (based on Saracevic, 2005), and criteria for evaluation
 * a. Information representations, metadata and surrogates used in the DL
 * i. Task appropriateness
 * ii. Usability
 * iii. User satisfaction
 * b. Particular tools available in the DL
 * i. Index, search, and output features
 * ii. Navigation, browsing
 * iii. Failures in functionality or usability
 * iv. User satisfaction
 * c. Particular services offered by the DL
 * i. Collection quality
 * ii. Retrieval performance (recall, precision)
 * iii. Reliability
 * iv. Human-intermediated services (e.g., reference services)
 * v. User satisfaction with individual services or with collection of services
 * d. User behaviors when interacting with the DL (may or may not be evaluative)
 * i. Information seeking/searching behaviors
 * ii. Use of information retrieved
 * iii. Work patterns

 * 5. "All efforts to design, implement, and evaluate digital libraries must be rooted in the information needs, characteristics, and contexts of the people who will or may use those libraries." Marchionini, Plaisant, & Komlodi, 2003, p.1

Questions that may be asked during an evaluation/user study

 * 1. Frame the study questions based on the decisions that must be made about the DL's functions/processes (Reeves et al., 2003)
 * a. Focus on those questions that are most important for making the decisions that are most important
 * b. Focus on impacts of DL functions/services (Marchionini, Plaisant, & Komlodi, 2003)
 * i. What types of impacts are there? On whom?
 * ii. Who and what influence those impacts?

 * 2. Formative versus summative evaluation
 * a. Formative evaluation focused on decisions about how to modify/change the DL's functions/services
 * b. Summative evaluation focused decisions about the worth or value of the DL's functions/services

Stages/steps in the evaluation/research process

 * 1. Develop an evaluation/research plan (Chowdhury & Chowdhury, 2003; Reeves et al., 2003)
 * a. Clarify the decisions to be addressed and the questions they generate
 * b. Identify appropriate evaluation methods, including sampling procedures, data collection procedures and data analysis procedures
 * c. Carry out the evaluation/research plan
 * d. Report the results to appropriate stakeholders
 * i. Primarily the decision makers
 * ii. Also other constituencies of the DL

 * 2. Review of an example evaluation, in terms of these steps
 * a. Possible example evaluations:
 * b. Bishop, A.P. (1998, December). Measuring access, use, and success in digital libraries. The Journal of Electronic Publishing, 4(2). Retrieved February 8, 2006, from www.press.umich.edu/jep/04-02/bishop.html.
 * c. Marchionini, G. (2000). Evaluating digital libraries: A longitudinal and multifaceted view. Library Trends, 49(2), 304-333.

Evaluation design strategies

 * 1. Naturalistic studies
 * a. For some evaluation studies, it is critical to conduct them in a natural or naturalistic setting/context
 * b. The constraints of the setting usually imply that fewer experimental controls can be applied to the study design
 * c. Usually, the evaluator will need to take into account aspects of the setting as part of the data collected for the evaluation study

 * 2. Experiments
 * a. Usually conducted in a lab setting, or a setting in which control over the conditions of the evaluation study can be exerted
 * b. The researcher attempts to control all the potential effects on the results of the study, other than those effects being intentionally manipulated as the focus of the evaluation

 * c. Some important concepts in designing an experiment
 * i. Randomization is a key tool for control: random sampling and/or random assignment to treatment and control groups
 * ii. Variables: the researcher will manipulate the independent variables (e.g., whether the DL has a particular feature or not) and will evaluate the outcomes based on the dependent variable
 * iii. The design may be a within-subjects design (where each participant interacts with all the variations of the independent variables, and so comparisons on the dependent variable are made "within" each subject's performance) or a between-subjects design (where each participant interacts with only one version of the system and comparisons are made "between" groups of subjects)

 * 3. Avoiding the effects of researcher bias

 * a. It's easy for a researcher's biases to influence the design of a study and, thus, its outcomes
 * i. Identify your biases
 * ii. Ensure that your study design and procedures will allow you to avoid any influence on the study outcomes

Data collection and measurement methods

 * 1. Collecting data from people requires ethical treatment of those people as study participants
 * a. Each institution will require review of the research proposal by an Institutional Review Board that verifies that study participants are being treated ethically

 * 2. Observation of user behaviors, including transaction logs
 * a. To see what the user is doing as he or she interacts with the system
 * b. Observation of work (e.g., via contextual inquiry)
 * i. Special kind of interview
 * ii. Observe person while performing task to be supported
 * iii. Interrupt with questions about how and why, as needed
 * c. Think-aloud protocols
 * i. During DL use, the participant is asked to verbalize their thought processes
 * ii. Allows you to observe "unobservable" cognitive behaviors
 * iii. Usually videotaped or audiotaped
 * d. Indirect observation of work
 * i. Logging and metering techniques embedded in the software of the current system or intermediate versions
 * e. Diaries
 * i. For detailed descriptions of tasks
 * How much time they take
 * Sequential dependencies between tasks
 * ii. Allows observation over longer periods of time than contextual inquiry interviews

 * 3. Interviews and focus groups
 * a. Augmenting other data collection methods, or on their own
 * b. Uses during DL evaluation
 * i. For identifying problems in the DL design
 * ii. For additional features needed in the DL
 * iii. For other improvements in the DL which the user can suggest
 * c. Individual interviews or group interview (focus groups)
 * i. Focus groups require a skilled facilitator

 * 4. Questionnaires
 * a. Surveys: typically one item/question per construct
 * b. Measures: intended to measure constructs that are not directly observable and not easily measured with a single item
 * i. The more subjective the construct, the more likely that you will need a multiple-item measure for it
 * Find a measure in the literature, rather than developing your own
 * c. Print vs. online administration
 * i. Print allows people to annotate (can be good or bad)
 * ii. Online eliminates the need for a separate data entry step

Study sample: Who should be participants in your evaluation study?

 * 1. Define the population of interest
 * a. Current users or a subset of them
 * b. Potential audiences (who are not current users)

 * 2. Consider sample size
 * a. Usually a tradeoff between small sample (cheaper) and generalizability
 * b. Intensive versus extensive studies
 * i. Intensive studies: to thoroughly understand a phenomenon within its context
 * ii. Extensive studies: to understand the extent of a phenomenon within a population

 * 3. Develop a sampling plan
 * a. Random sampling
 * i. Supports statistical inferences to the population
 * ii. Identify a population to which you want to generalize your findings
 * iii. Enumerate the population
 * iv. Draw a random sample
 * v. Problem: enumerating the entire population
 * Not necessarily problematic, but often is

 * 4. Other methods of sampling
 * a. Quota sampling, purposive sampling, accidental/convenience sampling
 * b. Strive for representativeness
 * i. In range, as well as central tendency

 * 5. Develop a plan for recruiting the sample you want
 * a. May need to offer incentives

Analysis and interpretation of data

 * 1. Reporting the results and interpreting the results are two distinct steps

 * 2. Interpretation should address the questions, "What do the results mean? How should they be understood?"

 * 3. All results must be interpreted in the context of:
 * a. Prior empirical work and relevant theoretical frameworks Situation
 * i. What is happening in the particular situation in which the study was done?
 * b. Weaknesses in the research
 * i. Measurements: level of reliability; validity
 * ii. Design: attrition, external events; internal and external threats to validity
 * iii. Analysis method: assumptions violated
 * c. Recommend particular actions, based on the interpretation of the results

Assigned readings for students

 * i. Nicholson, Scott. (2004). A conceptual framework for the holistic measurement and cumulative evaluation of library services. Journal of Documentation, 60(2), 164-182.
 * ii. Reeves, Thomas, Apedoe, Xornam, & Hee Woo, Young. (2003). Evaluating digital libraries: A user-friendly guide. University Corporation for Atmospheric Research; National Science Digital Library. Retrieved 3/1/2007 from http://www.dpc.ucar.edu/projects/evalbook/EvaluatingDigitalLibraries.pdf.
 * a. Chapter 1, Why evaluate? (p.1-6)
 * b. Chapter 2, Evaluating planning (p.7-21)

Recommended background reading for instructor

 * i. Chowdhury, G.G., & Chowdhury, S. (2003). Digital library evaluation. In Introduction to Digital Libraries. London: Facet Publishing, 267-283.
 * ii. Frechtling, J. (2002). The 2002 User Friendly Handbook for Project Evaluation. National Science Foundation, Directorate for Education & Human Resources, Division of Research, Evaluation, and Communication.
 * iii. Marchionini, G., Plaisant, C., & Komlodi, A. (2003). The people in digital libraries: Multifaceted approaches to assessing needs and impact. In Bishop, A.P., Van House, N.A., & Buttenfield, B.P. (eds.), Digital Library Use: Social Practice in Design and Evaluation. Cambridge, MA: MIT Press, 119-160.
 * iv. Reeves, Thomas, Apedoe, Xornam, & Hee Woo, Young. (2003). Evaluating digital libraries: A user-friendly guide. University Corporation for Atmospheric Research; National Science Digital Library. Retrieved 3/1/2007 from http://www.dpc.ucar.edu/projects/evalbook/EvaluatingDigitalLibraries.pdf.
 * v. Saracevic, T. (2000). Digital library evaluation: Toward evolution of concepts. Library Trends, 49(2), 350-369.

Additional Potential Readings

 * i. Bishop, A.P. (1998, December). Measuring access, use, and success in digital libraries. The Journal of Electronic Publishing, 4(2). Retrieved 2/8/2006 from www.press.umich.edu/jep/04-02/bishop.html.
 * ii. Bishop, A.P., Mehra, B., Bazzell, I., & Smith, C. (2003). Participatory action research and digital libraries: Reframing evaluation. In Bishop, A.P., Van House, N.A., & Buttenfield, B.P. (eds.), Digital Library Use: Social Practice in Design and Evaluation. Cambridge, MA: MIT Press, 161-189.
 * iii. Bollen, J. and R. Luce. (2002). Evaluation of digital library impact and user communities by analysis of usage patterns. D-Lib Magazine, 8(6) June 2002. Retrieved 3/1/2007 from http://www.dlib.org/dlib/june02/bollen/06bollen.html
 * iv. Bryan-Kinns, Nick & Blandford, Ann. (2000). A survey of user studies for digital libraries. RIDL Working Paper. Retrieved 3/1/2007 from http://www.cs.mdx.ac.uk/ridl/DLuser.pdf.
 * v. Choudhury, G.S.; Hobbs, B.; M Lorie, Flores, N.E. (2002, July/August). A framework for evaluating digital library service. D-Lib Magazine, 8(7/8). Retrieved 3/1/2007 from http://www.dlib.org/dlib/july02/choudhury/07choudhury.html.
 * vi. Marchionini, G. (2000). Evaluating digital libraries: A longitudinal and multifaceted view. Library Trends, 49(2), 304-333.
 * vii. Rieger, R., & Gay, G. (1999, June 15). Tools and Techniques in Evaluating Digital Imaging Projects. RLG DigiNews.
 * viii. Saracevic, Tefko (2005). How were digital libraries evaluated? Presented at Libraries in the Digital Age (LIDA), Dubrovnik and Mljet, Crotia, May 30-June 3. Retrieved 3/1/2007 from http://www.scils.rutgers.edu/~tefko/DL_evaluation_LIDA.pdf.
 * ix. Thong, J. (2002). Understanding user acceptance of digital libraries: What are the roles of interface characteristics, organizational context, and individual differences? International Journal of Human-Computer Studies, 57(3), 215-242.

Concept map

 * None

Exercises / Learning activities

 * 1. Analyze a DL evaluation report
 * a. Exercise 13.a, "Analyze a DL evaluation report," could be adapted to an in-class small-group discussion exercise. If so, the results of the each group's analysis could be reported orally or could be posted to a class wiki or discussion forum.

 * b. If used as an in-class exercise, assign the groups to read a particular evaluation report before class, to prepare for their in-class discussion and report.

 * c. Time requirements: 2 hours of preparation outside of class; 25-30 minutes for discussion in class; 20-30 minutes for report presentation in class, depending on the number of groups.

 * 2. Develop an evaluation plan
 * a. Based on a DL that is familiar to all the students in the class (e.g., flickr, MySpace, a music collection, the university's OPAC or a special collection that is well-known), have students work in small teams (3-4 people each) to develop a draft evaluation plan. They can play the role of an evaluation consulting firm, designing an evaluation study for their client, the DL managers.

 * b. Each plan should include the following:

 * i. The evaluation questions to be addressed
 * Stated briefly, in one sentence (preferably ending with ?)
 * ii. The sample to be included
 * How they will be selected
 * How they will be recruited
 * iii. The methods for data collection
 * The types of data to be collected, and how each pertains to the evaluation question
 * The procedures for collecting the needed data

 * c. Have each team of students present their plan to the class, as if it were an initial presentation to the client (the DL managers).

 * d. Time requirements: It is expected that the students will prepare their report after this module has been presented in class. Students should expect to spend 4-5 hours outside of class, preparing their reports. Each report should be presented in 7-10 minutes, during the next class session.

 * 3. Interview a digital librarian about evaluation
 * a. Note: This exercise is only possible if there are a number of robust local digital library projects, and the students will have access to their directors/administrators.

 * b. For this exercise, students should work in pairs; each pair will be assigned to investigate a particular digital library. Prior to the interview, each pair should read the available documentation on the digital library on which they're focused. Using the following interview guide, they should interview the director/administrator of the digital library.

 * c. Interview guide:
 * i. When was the digital library first established?
 * ii. What are the primary goals of the DL?
 * iii. In what ways do you evaluate whether you're achieving those goals?
 * iv. Do you evaluate any other aspects of the DL's operations? If so, how?
 * v. How and to whom are the evaluation results reported?

 * d. Each pair should write up a brief (1-3 page) summary of their interview findings. In addition, they should be prepared to orally report on the most interesting aspects of those findings at the next class session.

 * e. Time requirements, outside of class: 1-2 hours for preparatory reading; 1 hour for conducting the interview; 2-3 hours for writing up the interview report.
 * f. Time requirements, in class: 30-40 minutes for the class to discuss the findings from the interviews.

Evaluation of learning outcomes

 * 1. Analyze a digital library evaluation report
 * a. Using Saracevic's (2005) meta-analysis of digital library (DL) evaluations as a framework, evaluate an additional DL evaluation report. The report can be selected from the following :
 * i. Byrd, S., et al. (2001). Cost/benefit analysis for digital library projects: The Virginia Historical Inventory project (VHI). The Bottom Line: Managing Library Finances, 14(2), 65-75.
 * ii. Gambles, A. (2001). The HeadLine personal information environment: Evaluation Phase One. D-Lib Magazine, 7(3). www.dlib.org/dlib/march01/gambles/03gambles.html.
 * iii. Palmer, D., & Robinson, B. (2001). Agora: The hybrid library from a user's perspective. Ariadne, 26. www.ariadne.ac.uk/issue26/case-studies/intro.htm.
 * iv. Zhang, Y., Lee, K., & You, B.-J. (2001). Usage patterns of an electronic theses and dissertations system. Online Information Review, 25(6), 370-377.
 * v. List of possibilities; still need to be viewed.

 * b. Analyze the evaluation report in terms of the following aspects:
 * i. Construct for evaluation.
 * What was evaluated? What was actually meant by a "digital library"? What elements (components, parts, processes…) were involved in evaluation?
 * ii. Context of evaluation - selection of a goal, framework, viewpoint or level(s) of evaluation.
 * What was the basic approach or perspective? What was the level of evaluation? What was the objective(s)?
 * iii. Criteria reflecting performance as related to selected objectives.
 * What parameters of performance were concentrate[d] on? What dimension or characteristic [was] evaluated?
 * iv. Methodology for doing evaluation.
 * What measures and measuring instruments were used? What samples? What procedures were used for data collection? For data analysis?
 * v. Findings from evaluation studies
 * Only a single generalization is provided." (Saracevic, 2005, p.2-3)

 * c. Prepare a report (2-5 pages, single-spaced) summarizing the findings of your analysis.

 * d. The report should be evaluated in terms of its demonstration that the authors understood the DL evaluation conducted, its coverage of the five aspects of evaluations posed by Saracevic, its identification of strengths and weaknesses in the DL evaluation, and its clarity (organization, grammar, etc.).

 * e. Time requirements: approximately 6-8 hours outside of class.

 * 2. Develop an evaluation plan
 * a. Class exercise 10.b, "Develop an evaluation plan," could be adapted as a graded assignment. Each team would be expected to develop their plan over the week after the class's discussion of evaluation. If class time is available, the final plan can be presented orally; or, if preferred, the evaluation plans could be turned in as an evaluation proposal (2-4 pages).

 * b. The evaluation plans would be evaluated in terms of their completeness (were all the major components of an evaluation study addressed?), their feasibility (could the evaluation study actually be conducted, given reasonable resources?), and their clarity.

 * c. Time requirements: 6-8 hours outside of class, preparing and writing the evaluation plan.

Glossary

 * a. Between-subjects design: A research design in which "each research participant receives only one level of the independent variable" (Schmidt, 2000).
 * b. Dependent variable: "A variable that may, it is believed, be predicted by or caused by one or more other variables called independent variables." (U.S. Dept. of Justice, n.d.)
 * c. Evaluation: "An appraisal of the performance or functioning of a system, or part thereof, in relation to some objective(s)" Saracevic, 2000, p.359
 * d. Formative evaluation: An evaluation that is intended to "strengthen or improve the object being evaluated. Formative evaluations are used to improve [information systems] while they are still under development." (Trochim, 2001, p.347)
 * e. Independent variable: "A variable that may, it is believed, predict or cause fluctuation in an dependent variable." (U.S. Dept. of Justice, n.d.)
 * f. Research design: "A plan of what data to gather, from whom, how and when to collect the data, and how to analyze the data obtained." (U.S. Dept. of Justice, n.d.)
 * g. Sample: "The actual units you select to participate in your study." (Trochim, 2001, p.351)
 * h. Stakeholders: "People who have a vested interest in the success of the project or are involved in the implementation of the project." (California State University, Monterey Bay, n.d.)
 * i. Summative evaluation: An evaluation that "examine[s] the effects or outcomes of [an information system]." (Trochim, 2001, p.352)
 * j. Within-subjects design: A research design in which "each research participant provides data for all the levels of the independent variable" (Schmidt, 2000).

 * References for glossary:
 * a. Saracevic, T. (2000). Digital library evaluation: Toward evolution of concepts. Library Trends, 49(2), 350-369.
 * b. Trochim, W.M.K. (2001). The Research Methods Knowledge Base. Second ed. Cincinnati, OH: Atomic Dog Publishing.
 * c. U.S. Dept. of Justice, Office of Justice Programs, Bureau of Justice Assistance, Center for Program Evaluation. (n.d.). Glossary. Retrieved 5/23/2007 from http://www.ojp.usdoj.gov/BJA/evaluation/glossary/glossary_r.htm.

Additional useful links

 * None

Contributors

 * a. Developers:
 * i Barbara Wildemuth (UNC)
 * b. Evaluators:
 * i. Jeff Pomerantz (UNC)
 * ii. Sanghee Oh (UNC)
 * iii. Mike Christel (CMU)
 * iv. Flora McMartin (Broad-based Knowledge)
 * v. Liz Liddy (Syracuse)


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User talk:François Dongier

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User talk:Egm6321.f09.team7.mm

 Hello Egm6321.f09.team7.mm, 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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 * 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! --Mu301Bot 17:07, 25 September 2009 (UTC)

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text:
User:Egm6321.f09.Team1.sallstrom/HW2

= Problem on page 7-1 =

= Problems on pages 8-3 and 8-4 = Given the equation

y' + \frac{1}{x} y = x^2 $$
 * 1) Show that the solution for the integrating factor is $$h(x)=x$$
 * 2) Solve for y

Solution
Using the form

h y' + h' y = (hy)' = h b $$ and substituting

h(x) = e^{g(x)} $$ We have that

h y' + h' y = h \left( y' + g' y \right) = h \left( y' + \frac{1}{x} y \right) $$ So

g'(x) = \frac{1}{x} \quad \Rightarrow g(x) = \ln(x) + C $$ We can assume that $$C=0$$ without changing the end result. This yields

h(x) = e^{ln(x)} = x $$ Now, integrating

(hy)' = h b $$ yields the final solution

y(x) = \frac{1}{h(x)} \int^x h(s) b(s) ds = \frac{1}{x} \int^x s^3 ds = \frac{1}{x} \left[ \frac{x^4}{4} + C \right] = \frac{x^3}{4} + \frac{C}{x} $$ where $$C$$ is a constant.

= Problem on page 9-2 = With the definition

\bar b(x) := \int^x b(s) ds $$ and the equation

\bar b(x) y'(x) + \left[a(x) y(x) + k \right] = 0 $$ Let
 * $$ a(x) = x^4 \frac{}{}$$
 * $$ b(x) = x \Rightarrow \bar b(x) = \frac{1}{2} x^2 $$
 * $$ k = 10 \frac{}{} $$

This yields the equation

\frac{1}{2} x^2 y' + \left[ x^4 y + 10 \right] = 0 $$ Show that the above equation has an exact solution

Solution
Rewrite the given equation

y' + 2 x^2 y = -20 x^{-2} $$ Then multiply by $$h(x) = e^{g(x)}$$

h(x) y' + h(x) 2 x^2 y = -20 h(x) x^{-2} $$ Note that $$h'(x) = g'(x) h(x)$$, and substitute

h(x) y' + \frac{h'(x)}{g'(x)}2 x^2 y = -20 h(x) x^{-2} $$ Now, we use

g'(x) = 2 x^2 \Rightarrow g(x) = \frac{2}{3} x^3 $$ So that

h(x) y' + h'(x) y = -20 h(x) x^{-2} $$

\left( h(x) y \right)'= -20 h(x) x^{-2} $$ Now, integrate both sides

y = -\frac{1}{h(x)} \int^x 20 h(s) s^{-2} ds $$ Substitute in $$h(x)$$

y = -\frac{1}{\exp(2 x^3/3)} \int^x 20 \exp(2 s^3/3) s^{-2} ds $$

title: User:Egm6321.f09.Team1.sallstrom/HW2


text:
User:Egm6321.f09.Team1.sallstrom/HW3

=Problem 1: Solving for H(x,y) to ensure Exactness=

Given
A function is given by
 * {| style="width:100%" border="0" align="left"

F = x^{m}y^{n} \lbrack \sqrt{x}y''+2xy'+3y \rbrack = 0 $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

h(x,y) = x^{m}y^{n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Find
Find (m,n) such that Eq 1 is exact.

Solution
The first criteria for exactness for a second order non-liner ODE is that it is of the form
 * {| style="width:100%" border="0" align="left"

F(x,y,p) = f(x,y,p)y'' + g(x,y,p) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
 * }
 * }

where

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

p = y' $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 4)
 * }
 * }

Comparing Eq. 1 with the form of Eq. 3, the functions f and g can be defined as

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

f(x,y,p)=x^{0.5+n}y^{m} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 5)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

g(x,y,p)=2x^{n+1}y^{m}p+3x^{n}y^{m+1} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 6)
 * }
 * }

Therefore, the first conidition for exactness is met.

The second condition for exactness is given by the following equations
 * {| style="width:100%" border="0" align="left"

f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp} - g_{y} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 7)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

f_{xp} + pf_{yp} + 2f_{y} = g_{pp} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 8)
 * }
 * }

Analyzing Eq. 8 first, it should be noted that most of the terms drop out of the equation because the given f function (Eq. 5) is not a function of p. Therefore all derivatives of the function f wrt. p are zero. In addition to this, $$g_p$$ is constant wrt. $$p$$ and hence $$g_{pp}$$ is zero. Eq. 8 is simplified below

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

2f_y=g_{pp}=0 $$ $$ Taking the derivative of Eq. 5 and substituting into Eq. 9 yields
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 9)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

2\left[x^{0.5+n}my^{m-1}\right]=0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 10)
 * }
 * }

Eq. 11 must be true for the exactness condition to be met. Therefore, m=0 satisfies this condition.
 * {| style="width:100%" border="0" align="left"

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

m = 0 $$ $$
 * $$\displaystyle (Eq. 11)
 * }
 * }

Now, the second condition of exactness must be satisfied, specifially Eq. 7. Because m=0, the function f and g can be rewritten yielding
 * {| style="width:100%" border="0" align="left"

f=x^{0.5+n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 12)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

g=2x^{n=1}p+3x^{n}y $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 13)
 * }
 * }

The following derivatives of f and g are calculated in order to plug into Eq. 7.
 * {| style="width:100%" border="0" align="left"

\begin{array}{*{20}l} {f_{xx} = (n+0.5)(n-0.5)x^{-1.5+m}} \\ {f_{xy} = 0} \\ {f_{yy} = 0} \\ {g_{xp} = 2(n+1)x^{n}} \\ {g_{yp} = 0} \\ {g_y=3x^{n}} \\
 * $$\displaystyle
 * $$\displaystyle

\end{array} $$

 * }
 * }

Substiuting into Eq 7 and solving for n to ensure exactness yields
 * {| style="width:100%" border="0" align="left"

(n+0.5)(n-0.5)x^{-1.5+n} = 2(n+1)x^{n}-3x^{n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 14)
 * }
 * }

The LHS of equation 14 must be zero, since there is nothing to balance a $$x^{-1.5+n}$$ on the right hands side. The LHS is zero if $$n= \pm 0.5$$. If the LHS is zero, so must the RHS. It is zero only if
 * {| style="width:100%" border="0" align="left"

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

n = \frac{1}{2} $$ $$
 * $$\displaystyle (Eq. 15)
 * }
 * }

=Problem 2: Solving a L1-ODE=

Given
A function is given by
 * {| style="width:100%" border="0" align="left"

\phi(x,y,p) = xp+(2x^{\frac{3}{2}}-1)y - k_1 = k_2 $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

p = y' $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 2)
 * }
 * }

Find
Solve Eq. 1 for y(x).

Solution
Rewriting Eq. 1 by dividing through by x

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

y'+(2x^{0.5}-x^{-1})y = \frac{k_1 + k_2}{x} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 3)
 * }
 * }

In order to solve this problem, it is desired to form the LHS of the equation in the form of the product rule. The procedure for this is to multiply the entire equation through exp(f(x)) resulting in

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

e^{f(x)}y'+(2x^{0.5}-x^{-1})e^{f(x)}y=\frac{k_1+k_2}{x}e^{f(x)} $$ $$ The LHS needs to be solved such that it is in the form
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 4)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

(e^{f(x)}y)'=e^{f(x)}y'+f'(x)e^{f(x)}y $$ $$ By comparison with the LHS Eq. 4, it can be shown that
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 5)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

f(x)' = 2x^{0.5}-x^{-1} $$ $$ Integrating yields
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 6)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

f(x) = \frac{4}{3}x^{1.5}-\ln(x) + C = \frac{4}{3}x^{1.5}-\ln(x) $$ $$ Where $$C$$ can be arbitrarily chosen, hence we choose $$C=0$$. Therefore plugging into exp(f(x)) results in
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 7)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

e^{f(x)} = \frac{1}{x}e^{1.33x^{1.5}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 8)
 * }
 * }

Going back to Eq. 5 and integrating both sides results in
 * {| style="width:100%" border="0" align="left"

e^{f(x)}y=(k_1+k_2)\int^x\frac{1}{x}e^{f(x)}dx $$ $$ A change of variable will now be used to reduce confusion within the integral
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 9)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

e^{f(x)}y=(k_1+k_2)\int^x\frac{1}{a}e^{f(a)}dx $$ $$ Finally, the function y is solved for by dividing by exp(f(x)) and substituing for f(x) and f(a).
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 10)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

 * style="width:30%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

y=xe^{-1.33x^{1.5}}(k_1+k_2)\int^x\frac{1}{\xi} e^{\frac{4}{3}\xi^{1.5}}d\xi $$ $$
 *
$$\displaystyle (Eq. 11)
 * }
 * }

=Problem 3: Determining Mathamatical Structure yield an exact L2_ODE_VC=

Given
The function is a class of exact Linear Second Order ODE with varying coefficents.

Find
In order to invent new exact L2-ODE_VC, find the mathamatical structure of phi that yields the above class.

Solution
The form of a L2_ODE_VC is given by

The form of a 2N-ODE is given by
 * {| style="width:100%" border="0" align="left"

F(x,y,p,y) = P(x) y + Q(x) p + R(x) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 1)
 * }
 * }

where

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

p = y' $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 2)
 * }
 * }

The first exactness condition for a L2_ODE_VC is

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

F(x,y,p,y) = \phi_x + \phi_y p + \phi_p y = \frac{d\phi(x,y,p)}{dx} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 3)
 * }
 * }

By comparing equation 3 and 5, we see that

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

\phi_p = P(x) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 4)
 * }
 * }

Integrating yields

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

\phi = P(x) p + C(x,y) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 5)
 * }
 * }

Then

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

\phi_x = P'(x) p + \frac{\partial C(x,y)}{\partial x} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 6)
 * }
 * }

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

\phi_y = \frac{\partial C(x,y)}{\partial y} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 7)
 * }
 * }

Substituting these into Eq. 3 yields

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

F(x,y,p,y) = P(x) y + \underbrace{\left(P'(x) + \frac{\partial C}{\partial y}\right)}_{Q(x)} p + \underbrace{\frac{\partial C}{\partial x}}_{R(x) y} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 8)
 * }
 * }

We notice by comparing Eq. 8 with Eq. 1 that

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

\frac{\partial C}{\partial x} = R(x) y $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 9)
 * }
 * }

Integrating the above yields

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

C = y\underbrace{\int_0^x R(\xi) d\xi}_{T(x)} + K(y) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 10)
 * }
 * }

Substitute this into Eq. 5

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

\phi(x,y,p) = P(x) p + T(x) y + K(y) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 11)
 * }
 * }

The second exactness condition is
 * {| style="width:100%" border="0" align="left"

f_0 - \frac{df_1}{dx} + \frac{d^2 f_2}{dx} = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 12)
 * }
 * }

where

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

f_i = \frac{\partial F}{\partial y^{i}} = \frac{\partial}{\partial y^{i}}\left(\frac{d F}{dx}\right) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 13)
 * }
 * }

We have

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

F=\frac{d\phi}{dx} = P(x) y'' + \left( P'(x) + T(x) + K'(y)\right)p + T'(x) y $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 14)
 * }
 * }

Then

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

f_0 = \frac{\partial F}{\partial y} = K'(y) + T'(x) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 14)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\frac{df_1}{dx} = \frac{d}{dx}\left(\frac{\partial F}{\partial p}\right) = \frac{d}{dx}\left( P'(x) + T(x) + K'(y)\right) = P(x) + T'(x) + K(y) p $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 14)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\frac{d^2 f_2}{dx^2} = \frac{d^2 }{dx^2} \left(\frac{\partial F}{\partial y}\right) = \frac{d^2 P(x)}{dx^2} = P(x) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 14)
 * }
 * }

Putting these together in Eq. 12 yields
 * {| style="width:100%" border="0" align="left"

K'(y) + T'(x) - P(x) - T'(x) - K(y) p + P(x) = K'(y) - K(y) p = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 14)
 * }
 * }

=Problem 4: Solving for N=1 form of Exactness Conditions=

Given
A non linear 1st Order ODE is defined by
 * {| style="width:100%" border="0" align="left""

F(x,y,y') = 0= \frac{d\phi}{dx}(x,y) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 1)
 * }
 * }

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

1) f_o ( \phi) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 2)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

2) f_1 = \phi_y $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 3)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

3) f_o - \frac{df_1}{dx} = 0 \Leftrightarrow \phi_{xy}=\phi_{yx} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 4)
 * }
 * }

Solution
Starting with the general definition
 * {| style="width:100%" border="0" align="left"

f_n = \frac{\partial F(x,y,p)}{\partial y^{n}} $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 5)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

p=y' $$ $$ Solving Part 1:
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 6)
 * }
 * }

Pluging n=0 into Eq. 5 yields
 * {| style="width:100%" border="0" align="left"

f_o=\frac{\partial F(x,y)}{\partial y} $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 7)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

F(x,y,p)=\frac{d \phi(x,y)}{dx} $$ $$ Substituting Eq. 8 into Eq. 7 results in
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 8)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

f_o=\frac {\partial}{\partial y} \left[\frac{d \phi}{d x}\right]= \frac{\partial}{\partial y}\left[\frac{\partial \phi}{\partial x}+\frac{\partial \phi}{\partial y}\frac{d y}{d x}\right]=\phi_{xy} + \phi_{yy}\frac{d y}{d x} $$ $$ resulting in
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 9)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

f_o=\phi_{xy} + \phi_{yy}p $$ $$
 *
$$\displaystyle (Eq. 10)
 * }
 * }

Solving Part 2:

Substituting n=1 into Eq. 5 yields
 * {| style="width:100%" border="0" align="left"

f_1 = \frac{\partial F}{\partial y^{(1)}} = \frac{\partial F}{\partial p} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 11)
 * }
 * }

From Part 1, it can be seen that
 * {| style="width:100%" border="0" align="left"

F(x,y,p) = \frac{d\phi(x,y)}{dx} = \frac {\partial \phi}{\partial x} + \frac{\partial \phi}{\partial y}\frac{dy}{dx}=\phi_{x}+\phi_yp $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 12)
 * }
 * }

Subsituting Eq. 12 into Eq. 11 yields and differentiating
 * {| style="width:100%" border="0" align="left"

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

f_1=\phi_y $$ $$
 *
$$\displaystyle (Eq. 13)
 * }
 * }

Solving Part 3: The following equation defines the 2nd Exactness condition
 * {| style="width:100%" border="0" align="left"

f_o- \frac {df_1}{dx}=0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 14)
 * }
 * }

Taking the derivative of the f_1 term yields
 * {| style="width:100%" border="0" align="left"

\frac{df_1}{dx} = \frac{\partial \phi_y}{\partial x} + \frac {\partial \phi_y}{\partial y}\frac {dy}{dx} = \phi_{yx} + \phi_{yy}p $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 15)
 * }
 * }

Substituting Eq. 10 and 15 into Eq. 14 yields
 * {| style="width:100%" border="0" align="left"

\phi_{xy} + \phi_{yy} p - \phi_{yx} - \phi_{yy} p=0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 16)
 * }
 * }

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

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\phi_{xy} = \phi_{yx} $$ $$
 *
$$\displaystyle (Eq. 17)
 * }
 * }

=Problem 5: Solving for N=2 form of Exactness Conditions=

Given
A non linear 2nd Order ODE is defined by
 * {| style="width:100%" border="0" align="left""

F(x,y,y',y'') = 0= \frac{d\phi}{dx}(x,y,y') $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 1)
 * }
 * }

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

1) f_1 = \frac{df_2}{dx} + \phi_y $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 2)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

2) f_o = \frac{d\phi_y}{dx} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 3)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

3) f_o - \frac{df_1}{dx} + \frac{d^2f_2}{dx^2}= 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 4)
 * }
 * }

Secondly, relate Eq 4 to previously derived equations for exactness given by
 * {| style="width:100%" border="0" align="left"

f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp} - g_{y} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 5)
 * }
 * }

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

f_{xp} + pf_{yp} + 2f_{y} = g_{pp} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 6)
 * }
 * }

Solution
From the basic definition of the function
 * {| style="width:100%" border="0" align="left""

F(x,y,p,q) = \frac {d}{dx}\phi(x,y,p) = \frac{\partial \phi}{\partial x}+\frac{\partial \phi}{\partial y}\frac{d y}{d x}+\frac{\partial \phi}{\partial p}\frac{d p}{d x} = \phi_x + \phi_y p + \phi_p q $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 7)
 * }
 * }
 * {| style="width:100%" border="0" align="left""

p=y' $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 8)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

q = y'' $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 9)
 * }
 * }

The second condition of exactness is given by
 * {| style="width:100%" border="0" align="left"

f_o - \frac{df_1}{dx} + \frac{d^2f_2}{dx^2}-...+(-1)^n\frac{d^{n}}{dx^{n}}f_n=0 $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 10)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

f_i :=\frac{\partial F}{\partial y^{i}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 11)
 * }
 * }

Part 1: Solving for $$f_1$$

By substituting Eq. 7 into Eq. 11 using i = 1, then differentiating yields
 * {| style="width:100%" border="0" align="left""

f_1 = \frac{\partial F}{\partial p} = \phi_{xp} + \phi_{yp} p + \phi_y + \phi_{pp} q $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 12)
 * }
 * }

The same procedure is follow substituting Eq. 7 into Eq. 11 using i = 2, then differentiating yields
 * {| style="width:100%" border="0" align="left""

f_2 = \frac{\partial F}{\partial q} = \phi_p $$ $$ Differentiating Eq. 13 yields
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 13)
 * }
 * }
 * {| style="width:100%" border="0" align="left""

\frac{d f_2}{dx} = \phi_{px} + \phi_{py} \frac{dy}{dx} + \phi_{pp} \frac{dp}{dx} = \phi_{px} + \phi_{py} p + \phi_{pp} q = f_1 - \phi_y $$ $$ which can be rewritten
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 14)
 * }
 * }

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

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

f_1 = \frac{d f_2}{dx} + \phi_y $$ $$
 *
$$\displaystyle (Eq. 15)
 * }
 * }

Part 2: Solving for $$f_0$$

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

f_0 = \frac{\partial }{\partial y} \left[\frac{d \phi}{d x} \right] = \frac{\partial }{\partial y} \left[\frac{\partial \phi}{\partial x} + \frac{\partial \phi}{\partial y} p + \frac{\partial \phi}{\partial p} q\right] = \frac{\partial^2 \phi}{\partial x \partial y} + \frac{\partial^2 \phi}{\partial y^2} p + \frac{\partial^2 \phi}{\partial p \partial y} q = \frac{\partial \phi_y}{\partial x} + \frac{\partial \phi_y}{\partial y} p + \frac{\partial \phi_y}{\partial p} q = \frac{d \phi_y}{dx} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 16)
 * }
 * }

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

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

f_0 = \frac{d \phi_y}{dx} $$ $$
 *
$$\displaystyle (Eq. 17)
 * }
 * }

Part 3: Proving Exactness Condition

Eq. 4 can be written
 * {| style="width:100%" border="0" align="left""

f_0 - \frac{d}{dx}\left[ f_1 - \frac{d f_2}{dx}\right] $$ $$ Using Eq. 2, we can rewrite this as
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 18)
 * }
 * }
 * {| style="width:100%" border="0" align="left""

f_0 - \frac{d}{dx}\left[\phi_y\right] $$ $$ Using Eq. 3, this becomes
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 19)
 * }
 * }
 * {| style="width:100%" border="0" align="left""

\frac{d\phi_y}{dx} - \frac{d}{dx}\left[ \phi_y\right] = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 20)
 * }
 * }

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

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

0=0 $$ $$
 *
$$\displaystyle (Eq. 21)
 * }
 * }

'''Part 4: Relating Eq. 4 to Eq. 5 and 6'''

First, we can write $$F(x,y,p,q)$$ in terms of $$f(x,y,p)$$ and $$g(x,y,p)$$, by rewriting equation 7
 * {| style="width:100%" border="0" align="left""

F(x,y,p,q) = \underbrace{\phi_p}_{f(x,y,p)} q + \underbrace{\phi_x + p \phi_y}_{g(x,y,p)} = f y'' + g $$ $$ Next, we want to put $$f_0$$, $$f_1$$, and $$f_2$$ in terms of $$f$$ and $$g$$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 22)
 * }
 * }
 * {| style="width:100%" border="0" align="left""

f_0 = \frac{\partial F(x,y,p,q)}{\partial y} = f_y y'' + g_y $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 23)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

f_1 = \frac{\partial F(x,y,p,q)}{\partial p} = f_p y'' + g_p $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 24)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

f_2 = \frac{\partial F(x,y,p,q)}{\partial q} = f $$ $$ Next, we need to differentiate $$f_1$$ and $$f_2$$ with respect to $$x$$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 25)
 * }
 * }
 * {| style="width:100%" border="0" align="left""

\frac{df_1}{dx} = \left( f_{px} + f_{py} p + f_{pp} q \right) q + f_{p} y''' + g_{px} + p g_{py} + q g_{pp} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 26)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

\frac{df_2}{dx} = f_x + f_y p + f_p q $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 27)
 * }
 * {| style="width:100%" border="0" align="left""
 * {| style="width:100%" border="0" align="left""

\frac{d^2f_2}{dx^2} = f_{xx} + p f_{xy} + q f_{xp}+ \left(f_{yx} + f_{yy} p + f_{yp} q \right) p + \left(f_y + f_{px} + p f_{py} + q f_{pp} \right) q + f_p y''' $$ $$ Now, we can substitute Eqs. 23, 26, and 28 into Eq. 4, which yields
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 28)
 * }
 * }
 * {| style="width:100%" border="0" align="left""

f_y y'' + g_y - g_{px} - p g_{py} - q g_{pp} +f_{xx} + p f_{xy} + q f_{xp} + \left(f_{yx} + f_{yy} p + f_{yp} q \right) p + f_y q = 0 $$ $$ Rearranging the terms yields
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 29)
 * }
 * }
 * {| style="width:100%" border="0" align="left""

f_{xx} + 2 p f_{xy} + p^2 f_{yy} - g_{px} - p g_{py} + g_y + \left(f_{xp} + p f_{yp} + 2 f_y - g_{pp} \right)q = 0 $$ $$ For the above equation to be valid for arbitrary $$q$$, both the following equations must be satisfied for Eq 30 to be satisfied
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 30)
 * }
 * }

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

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

f_{xx} + 2 p f_{xy} + p^2 f_{yy} = g_{px} + p g_{py} - g_y $$ $$
 *
$$\displaystyle (Eq. 31)
 * }
 * }

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

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

f_{xp} + p f_{yp} + 2 f_y = g_{pp} $$ $$
 *
$$\displaystyle (Eq. 32)
 * }
 * }

=Problem 6=

Given
The Legendre differential equation is given by
 * {| style="width:100%" border="0" align="left"

F = (1-x^2)y'' - 2xy' +n(n+1)y = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 1)
 * }
 * }

Find
First: Verify the exactness of this equation using the 2 exactness conditions given by:

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

1: f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp} - g_{y} $$ $$
 * $$\displaystyle
 *
$$\displaystyle (Eq. 2)

 * }
 * }

and

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

2: f_{xp} - pf_{yp} + 2f_{y} = g_{pp} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 3)
 * }
 * }

Second: Also veryify exactness by
 * {| style="width:100%" border="0" align="left"

f_0 - \frac {df_1}{dx} + \frac {d^2f_2}{dx^2}=0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 4)
 * }
 * }

If not exact, then see wether it can be made exact by
 * {| style="width:100%" border="0" align="left"

h(x,y) = x^{m}y^{n}. $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 5)
 * }
 * }

Solution
Part 1: By definition, the function must have the form

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

F = f(x,y,p)y'' + g(x,y,p) $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 6)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

p = y'. $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 7)
 * }
 * }

Therefore, it is seen that
 * {| style="width:100%" border="0" align="left"

f(x,y,p) = 1-x^2 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 8)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

g(x,y,p) = -2xp+n(n+1)y $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 9)
 * }
 * }

Taking the partial derivatives yields
 * {| style="width:100%" border="0" align="left"

\begin{array}{*{20}l} {f_x = -2x} & {f_{xx}  = -2} & {g_x  = -2p} & {g_{xp}  = -2}  \\ {} & {f_{xy} = 0} & {g_y  = n(n+1)} & {g_{yp}  = 0}  \\ {} & {f_{xp} = 0 } & {g_p  = -2x} & {g_{pp}  = 0 }  \\ {f_y = 0} & {f_{yy}  = 0} & {} & {}  \\ {} & {f_{yp} = 0} & {} & {}  \\ \end{array} $$
 * $$\displaystyle
 * $$\displaystyle

Subsituting into Eq 2 to verify exactness condition 1 yields
 * }
 * }

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

-2 + 0 + 0 = -2 + 0 - n(n+1) $$ $$
 *
$$\displaystyle (Eq. 10)
 * }
 * }

Subsituting into Eq 3 to verify exactness condition 2 yields
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

0 + 0 + 0 = 0 $$ $$ Currently, Eq 10 and 11 prove condition 2 is satisfied but condition 1 is not and therefore the equation is not exact. If n is a free variable, the function can be considered exact if Eq. 10 is solved for n. This results in
 *
$$\displaystyle (Eq. 11)
 * }
 * }

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

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

n_1 = 0, \quad n_2 = -1 $$ $$ and an exact function.
 *
$$\displaystyle (Eq. 12)
 * }
 * }

Part 2:

=Problem 7 - Linearity Proof=

Given
The two properies of linearity conisist of an additive and multiplicative property shown in the following equations respectively:

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

L(u+v) = L(u) + L(v) $$ $$ and
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 1)
 * }
 * }

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

L(\lambda u) = \lambda L(u) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 2)
 * }
 * }

Find
Show that these equations can be combined to form
 * {| style="width:100%" border="0" align="left"

L(\alpha u+ \beta v) = \alpha L(u) + \beta L(v) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 3)
 * }
 * }

Solution
Using Eq. 1, we can show that The first step is to define two new relationships using Eq. 2 with the constants alpha and beta shown below.

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

L(\alpha u + \beta v) = L(\alpha u) + L(\beta v) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 4)
 * }
 * }

Then, using Eq. 2 and Eq. 4, we can shot that
 * {| style="width:100%" border="0" align="left"

L(\alpha u) + L(\beta v) = \alpha L(u) + \beta L(v) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 5)
 * }
 * }

Combining Eq. 4 and 5 yields:
 * {| style="width:100%" border="0" align="left"

L(\alpha u + \beta v) = \alpha L(u) + \beta L(v) $$ $$
 * $$ \displaystyle
 * $$ \displaystyle
 *
$$\displaystyle (Eq. 6)
 * }
 * }

Hence

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\left.\begin{matrix} L(u + v) = L(u) + L(v) \\ L(\alpha u) = \alpha L(u) \end{matrix}\right\} \Rightarrow L(\alpha u + \beta v) = \alpha L(u) + \beta L(v) $$ $$
 *
$$\displaystyle (Eq. 7)
 * }
 * }

Next, with $$\alpha, \beta=1$$, Eq. 3 becomes

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

L(u + v) = L(u) + L(v) $$ $$
 * $$ \displaystyle
 * $$ \displaystyle
 *
$$\displaystyle (Eq. 8)
 * }
 * }

Which means that

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

L(\alpha u + \beta v) = \alpha L(u) + \beta L(v) \Rightarrow L(u + v) = L(u) + L(v) $$ $$
 *
$$\displaystyle (Eq. 9)
 * }
 * }

Next, with $$v = 0$$, Eq. 3 becomes

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

L(\alpha u) = \alpha L(u) $$ $$
 * $$ \displaystyle
 * $$ \displaystyle
 *
$$\displaystyle (Eq. 10)
 * }
 * }

Hence

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

L(\alpha u + \beta v) = \alpha L(u) + \beta L(v) \Rightarrow L(\alpha u) = \alpha L(u) $$ $$
 *
$$\displaystyle (Eq. 11)
 * }
 * }

Combining Eqs. 7, 9, and 11 yields:

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\left.\begin{matrix} L(u + v) = L(u) + L(v) \\ L(\alpha u) = \alpha L(u) \end{matrix}\right\} \Leftrightarrow L(\alpha u + \beta v) = \alpha L(u) + \beta L(v) $$ $$
 *
$$\displaystyle (Eq. 12)
 * }
 * }

= Problem 8: Shape Functions =

Given
The shape functions $$N_{J}^1$$, $$N_{J+1}^1$$, and $$N_{J}^2$$ are shown below for $$J=0$$. They are third order polynomials, and

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

\begin{matrix} N_{J}^1(J)                = 1 & N_{J}^2(J)                 = 0 & N_{J+1}^1(J)                 = 0 \\ \left[N_{J}^1(J)\right]'  = 0 & \left[N_{J}^2(J)\right]'   = 1 & \left[N_{J}^1(J)\right]'     = 0 \\ N_{J}^1(J+1)              = 0 & N_{J}^2(J+1)               = 0 & N_{J+1}^1(J+1)               = 1 \\ \left[N_{J}^1(J+1)\right]' = 0 & \left[N_{J}^2(J+1)\right]' = 0 & \left[N_{J+1}^1(J+1)\right]' = 0 \end{matrix} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 1)
 * }
 * }

Find
Plot $$N^2_{J+1}(x)$$ for $$ x \in [J, J+1]$$

Solution
We have

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

N_{J}^1(x)     = C_{11} + C_{21} x + C_{31} x^2 + C_{41} x^3 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 2)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

N_{J}^2(x)     = C_{12} + C_{22} x + C_{32} x^2 + C_{42} x^3 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 3)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

N_{J+1}^1(x)   = C_{13} + C_{23} x + C_{33} x^2 + C_{43} x^3 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 4)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

N_{J+1}^2(x)   = C_{14} + C_{24} x + C_{34} x^2 + C_{44} x^3 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 5)
 * }
 * }

Which must satisfy the conditions

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

\begin{matrix} N_{J}^1(J)                = 1 & N_{J}^2(J)                 = 0 & N_{J+1}^1(J)                 = 0 & N_{J+1}^2(J)                 = 0 \\ \left[N_{J}^1(J)\right]'  = 0 & \left[N_{J}^2(J)\right]'   = 1 & \left[N_{J}^1(J)\right]'     = 0 & \left[N_{J+1}^2(J)\right]'   = 0 \\ N_{J}^1(J+1)              = 0 & N_{J}^2(J+1)               = 0 & N_{J+1}^1(J+1)               = 1 & N_{J+1}^2(J+1)               = 0 \\ \left[N_{J}^1(J+1)\right]' = 0 & \left[N_{J}^2(J+1)\right]' = 0 & \left[N_{J+1}^1(J+1)\right]' = 0 & \left[N_{J+1}^2(J+1)\right]' = 1 \end{matrix} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 6)
 * }
 * }

This can be put in the equation system

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

\underbrace{ \left[ \begin{matrix} 1 & J  &   J^2   &    J^3    \\ 0 & 1  &   2 J   &   3 J^2   \\ 1 & J+1 & (J+1)^2 & (J+1)^2  \\ 0 & 1  & 2 (J+1) & 3 (J+1)^2 \end{matrix} \right] }_A \underbrace{ \left[ \begin{matrix} C_{11} & C_{12} & C_{13} & C_{14} \\ C_{21} & C_{22} & C_{23} & C_{24} \\ C_{31} & C_{32} & C_{33} & C_{34} \\ C_{41} & C_{42} & C_{43} & C_{44} \end{matrix} \right] }_C = \underbrace{ \left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right] }_I $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 7)
 * }
 * }

The coefficients can be solved for by multiplying both sides by $$A^{-1}$$. This yields

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

C = A^{-1} I = A^{-1} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 8)
 * }
 * }

Hence, the coefficients are

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

\underbrace{ \left[ \begin{matrix} C_{11} & C_{12} & C_{13} & C_{14} \\ C_{21} & C_{22} & C_{23} & C_{24} \\ C_{31} & C_{32} & C_{33} & C_{34} \\ C_{41} & C_{42} & C_{43} & C_{44} \end{matrix} \right] }_C = \underbrace{ \left[ \begin{matrix} -(2 J - 1) (J + 1)^2 &   -J (J + 1)^2 &  J^2 (2 J + 3) & -J^2 (J + 1) \\ 6 J (J + 1) & 3 J^2 + 4 J + 1 & -6  J (J + 1) &  J (3 J + 2) \\ - 6 J - 3 &      - 3 J - 2 &        6*J + 3 &    - 3 J - 1 \\ 2 &              1 &             -2 &            1 \end{matrix} \right] }_{A^{-1}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 9)
 * }
 * }

So, we can simplify equation 9 by assuming $$J=0$$. This only shifts the location of the functions along the x-axis, it does not change it's shape. The simplification yields the coefficients

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

C = \left[ \begin{matrix} 1 & 0 &  0 &  0 \\ 0 &  1 &  0 &  0 \\ -3 & -2 &  3 & -1 \\ 2 &  1 & -2 &  1 \end{matrix} \right] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 10)
 * }
 * }

Hence, with the simplification $$J=0$$

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

N_{J+1}^2(x) = -x^2 + x^3 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 11)
 * }
 * }

The plot of the above equation is shown below

= Problem 9: Finding solution to Homogeneous L2-ODE-VC =

Given
A Homogeneous L2-ODE-VC is given by
 * {| style="width:100%" border="0" align="left"

x^2y_{xx} - 2xy_{x} + 2y = 0. $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 1)
 * }
 * }

The second derivative wrt x has been found to be
 * {| style="width:100%" border="0" align="left"

y_{xx} = e^{-2t}(y_{tt} - y_t) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 2)
 * }
 * }

It should be noted that in order to solve for the second derivative, a transform of variable was used as shown in the following equation
 * {| style="width:100%" border="0" align="left"

x=e^{t} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 3)
 * }
 * }

Find
Determine the 3rd and 4th derivatives wrt x shown in the following equations:

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

y_{xxx} = e^{-3t}(y_{ttt} - 3y_{tt} + 2y_t) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 4)
 * }
 * }

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

y_{xxxx} = e^{-4t}(y_{tttt} - 6y_{ttt} + 11y_{tt} - 6y_t) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 5)
 * }
 * }

Solution
=Contributing Team Members= Egm6321.f09.Team1.AH 19:43, 4 October 2009 (UTC)

Egm6321.f09.Team1.sallstrom 21:34, 4 October 2009 (UTC)

title: User:Egm6321.f09.Team1.sallstrom/HW3


text:
Ask Me 3

Why Important Nearly half of all American adults, 90 million people, have difficulty understanding and using health information that they are given (Partnership for Clear Health Communication, 2007). This statistic is important because health literacy skills – the ability to read, understand and effectively use basic medical instructions and information – are linked with an individual’s health status (Partnership for Clear Health Communication, 2007); which means the lower your health literacy skills are, the more likely you are to become sick or have lower than average health. This is why clear communication between patients and health care providers is essential (Partnership for Clear Health Communication, 2007). The better the communication between the health care provider and their patients, the more likely it is that the patient will have a better understanding of what their health problems are and why they need to follow their physician’s advice, and in turn can be one step closer to becoming a healthier person. The Ask Me 3 program is designed to do just that - to help patients increase their health literacy and, as a result, also raise their health.

What Questions to Ask The Ask Me 3 program is centered on three important universal questions that were developed to help increase health literacy. These three questions are (Partnership for Clear Health Communication, 2007):

 * What is my main problem?
 * What do I need to do?
 * Why is it important for me to do this?

These questions should be asked when patients are seeing any health care providers or pharmacists, when preparing for a medical test or procedure or when picking up medicine so you they will become a more informed consumer of their healthcare (Partnership for Clear Health Communication, 2007).

Advantages of Ask Me 3 There are many important advantages to asking these questions, such as keeping patients safe when taking their medication at home, helping them understand how to get better and being to better understand what procedures may be performed on them (Partnership for Clear Health Communication, 2007). They may also be able to spend less money on healthcare by making fewer trips to their healthcare provider because main questions were answered during the first visit instead of during consequential visits.

Other Important Questions to Ask The Ask Me 3 questions are not the only important questions that patients should ask their health care provider or pharmacist. It is also important that any other questions they thought of on their own are answered. These questions can be used to supplement the Ask Me 3 questions.

Partnership for Clear Health Communication. (2007). Ask Me 3. Retrieved from http://www.npsf.org/askme3/

title: Ask Me 3


text: Category:Emerging Technologies

title: Category:Emerging Technologies