http://drorbn.net/api.php?action=feedcontributions&user=Dongwoo.kang&feedformat=atomDrorbn - User contributions [en]2024-03-29T01:30:21ZUser contributionsMediaWiki 1.21.1http://drorbn.net/index.php?title=12-26712-2672012-12-17T23:17:36Z<p>Dongwoo.kang: </p>
<hr />
<div>__NOEDITSECTION__<br />
__NOTOC__<br />
{{12-267/Navigation}}<br />
==Advanced Ordinary Differential Equations==<br />
===Department of Mathematics, University of Toronto, Fall 2012===<br />
<br />
{{12-267/Crucial Information}}<br />
<br />
===Text===<br />
Boyce and DiPrima, [http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002451.html Elementary Differential Equations and Boundary Value Problems] (current edition is 9th and 10th will be coming out shortly. Hopefully any late enough edition will do).<br />
<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
<br />
* Vitali Kapovitch's 2007 classes: [http://www.math.toronto.edu/vtk/267/ Spring], [http://www.math.toronto.edu/vtk/267fall07/ Fall].<br />
<br />
* Also previously taught by T. Bloom, C. Pugh, D. Remenik.<br />
<br />
* My {{Pensieve Link|Classes/12-267/|12-267 notebook}}.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[[User:Drorbn|Drorbn]] 06:36, 12 September 2012 (EDT): Material by [[User:Syjytg|Syjytg]] moved to [[12-267/Tuesday September 11 Notes]].<br />
<br />
[http://imgur.com/a/OSx1U#0 Summary of techniques to solve differential equations ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Existence_And_Uniqueness_Theorem Fundamental Theorem and Proof from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Derivation_of_Euler-Lagrange Derivation of Euler-Lagrange from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://graphics.ethz.ch/teaching/former/vc_master_06/Downloads/viscomp-varcalc_6.pdf Useful PDF: proof of Euler-Lagrange equation, explanation, examples ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
12-267 [http://math.hunter.cuny.edu/mbenders/cofv.pdf In-depth coverage of Calculus of Variations][[User:Simon1|Simon1]]<br />
<br />
12-267 [http://highered.mcgraw-hill.com/sites/dl/free/007063419x/392340/Calculus_of_Variations.pdf A good summary of Calculus of Variations]<br />
[[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/1vy11#0 All class notes from September 10th to October 5th] [[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/uLSlM Summary of Numerical Methods] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/index.php?title=Numerical_Methods Numerical Methods (wiki)] [[User:Twine|Twine]]<br />
<br />
12-267 [http://imgur.com/a/tliYg Summary of Chapter 3 from the Textbook on Constant Coefficient Second Order ODEs] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/images/e/e6/Geometric_Interpretation_of_Lagrange_Multiplier.pdf Geometric Interpretation of Lagrange Multiplier] [[User:Mathstudent|Mathstudent]]<br />
<br />
Past exams from [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2009] and [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2010] [[User:Twine|Twine]]<br />
<br />
Handwritten notes by [[User:Ktnd3|Ktnd3]]:<br />
<br />
* September: [[Media:Mat267_-_lecture_1(sep.10).PDF|10th]], [[Media:Mat267_-_lecture_2%28sep.11%29.PDF|11th]], [[Media:Mat267_-_lecture_3%28sep.14%29.PDF|14th]], [[Media:Mat267_-_lecture_4%28sep.17%29.PDF|17th]], [[Media:12-267%28lecture5%29.PDF|18th]], [[Media:12-267%28lecture6%29.PDF|21st]], [[Media:12-267%28lecture7%29.PDF|24th]], [[Media:12-267%28lecture8%29.PDF|25th]], [[Media:12-267%28lecture9%29.PDF|28th]]<br />
<br />
* October: [[Media:12-267%28lecture10%29.PDF|1st]], [[Media:12-267%28lecture11%29.PDF|2nd]], [[Media:12-267%28lecture12%29.PDF|5th]], [[Media:12-267%28lecture13%29.PDF|9th]] [[Media:12-267%28lecture14%29.PDF|12th]]<br />
<br />
[http://i.imgur.com/uTugV.jpg Quick guide: system of 1st order linear equations] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Help of Inverse Matrix [http://mathworld.wolfram.com/MatrixInverse.html Matrix Inverse] [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<br />
12-267 [http://imgur.com/a/50sRR All class notes from October 5th to October 30th] [[User:Simon1|Simon1]]<br />
<br />
[http://imgur.com/a/sZSYx#0 Quick guide: Power Series + ODE] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[https://lh4.googleusercontent.com/-V1WnvvzXRlE/TxTGwl_BBjI/AAAAAAAAE9k/yVuy1zRmnWs/Gatos-cute-cute-32-800x500.jpg Mirrors do actually flip top-to-bottom, depending on how you look at them] [[User:jonathanrlove|jonathanrlove]]<br />
<br />
[http://www.guardian.co.uk/notesandqueries/query/0,5753,-19877,00.html Actually, mirrors don't flip top-to-bottom *or* left-to-right] [[User:Twine|Twine]]<br />
<br />
[[12-267/Topic_List|Topics covered this semester]] - [[User:Twine|Twine]]<br />
<br />
(Summary of ODE) Hope it is Helpful.<br />
[[Media:12-267(ODE-Summary).pdf]] - [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<br />
Review in types of ODEs and etc.<br />
[[Media:12-267(Review_of_ODEs).pdf]] - [[User:Dongwoo.kang|Dongwoo.kang]]</div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-26712-2672012-12-17T07:56:36Z<p>Dongwoo.kang: </p>
<hr />
<div>__NOEDITSECTION__<br />
__NOTOC__<br />
{{12-267/Navigation}}<br />
==Advanced Ordinary Differential Equations==<br />
===Department of Mathematics, University of Toronto, Fall 2012===<br />
<br />
{{12-267/Crucial Information}}<br />
<br />
===Text===<br />
Boyce and DiPrima, [http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002451.html Elementary Differential Equations and Boundary Value Problems] (current edition is 9th and 10th will be coming out shortly. Hopefully any late enough edition will do).<br />
<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
<br />
* Vitali Kapovitch's 2007 classes: [http://www.math.toronto.edu/vtk/267/ Spring], [http://www.math.toronto.edu/vtk/267fall07/ Fall].<br />
<br />
* Also previously taught by T. Bloom, C. Pugh, D. Remenik.<br />
<br />
* My {{Pensieve Link|Classes/12-267/|12-267 notebook}}.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[[User:Drorbn|Drorbn]] 06:36, 12 September 2012 (EDT): Material by [[User:Syjytg|Syjytg]] moved to [[12-267/Tuesday September 11 Notes]].<br />
<br />
[http://imgur.com/a/OSx1U#0 Summary of techniques to solve differential equations ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Existence_And_Uniqueness_Theorem Fundamental Theorem and Proof from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Derivation_of_Euler-Lagrange Derivation of Euler-Lagrange from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://graphics.ethz.ch/teaching/former/vc_master_06/Downloads/viscomp-varcalc_6.pdf Useful PDF: proof of Euler-Lagrange equation, explanation, examples ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
12-267 [http://math.hunter.cuny.edu/mbenders/cofv.pdf In-depth coverage of Calculus of Variations][[User:Simon1|Simon1]]<br />
<br />
12-267 [http://highered.mcgraw-hill.com/sites/dl/free/007063419x/392340/Calculus_of_Variations.pdf A good summary of Calculus of Variations]<br />
[[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/1vy11#0 All class notes from September 10th to October 5th] [[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/uLSlM Summary of Numerical Methods] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/index.php?title=Numerical_Methods Numerical Methods (wiki)] [[User:Twine|Twine]]<br />
<br />
12-267 [http://imgur.com/a/tliYg Summary of Chapter 3 from the Textbook on Constant Coefficient Second Order ODEs] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/images/e/e6/Geometric_Interpretation_of_Lagrange_Multiplier.pdf Geometric Interpretation of Lagrange Multiplier] [[User:Mathstudent|Mathstudent]]<br />
<br />
Past exams from [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2009] and [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2010] [[User:Twine|Twine]]<br />
<br />
Handwritten notes by [[User:Ktnd3|Ktnd3]]:<br />
<br />
* September: [[Media:Mat267_-_lecture_1(sep.10).PDF|10th]], [[Media:Mat267_-_lecture_2%28sep.11%29.PDF|11th]], [[Media:Mat267_-_lecture_3%28sep.14%29.PDF|14th]], [[Media:Mat267_-_lecture_4%28sep.17%29.PDF|17th]], [[Media:12-267%28lecture5%29.PDF|18th]], [[Media:12-267%28lecture6%29.PDF|21st]], [[Media:12-267%28lecture7%29.PDF|24th]], [[Media:12-267%28lecture8%29.PDF|25th]], [[Media:12-267%28lecture9%29.PDF|28th]]<br />
<br />
* October: [[Media:12-267%28lecture10%29.PDF|1st]], [[Media:12-267%28lecture11%29.PDF|2nd]], [[Media:12-267%28lecture12%29.PDF|5th]], [[Media:12-267%28lecture13%29.PDF|9th]] [[Media:12-267%28lecture14%29.PDF|12th]]<br />
<br />
[http://i.imgur.com/uTugV.jpg Quick guide: system of 1st order linear equations] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Help of Inverse Matrix [http://mathworld.wolfram.com/MatrixInverse.html Matrix Inverse] [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<br />
12-267 [http://imgur.com/a/50sRR All class notes from October 5th to October 30th] [[User:Simon1|Simon1]]<br />
<br />
[http://imgur.com/a/sZSYx#0 Quick guide: Power Series + ODE] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[https://lh4.googleusercontent.com/-V1WnvvzXRlE/TxTGwl_BBjI/AAAAAAAAE9k/yVuy1zRmnWs/Gatos-cute-cute-32-800x500.jpg Mirrors do actually flip top-to-bottom, depending on how you look at them] [[User:jonathanrlove|jonathanrlove]]<br />
<br />
[http://www.guardian.co.uk/notesandqueries/query/0,5753,-19877,00.html Actually, mirrors don't flip top-to-bottom *or* left-to-right] [[User:Twine|Twine]]<br />
<br />
[[12-267/Topic_List|Topics covered this semester]] - [[User:Twine|Twine]]<br />
<br />
(Summary of ODE) Hope it is Helpful.<br />
[[Media:12-267(ODE-Summary).pdf]] - [[User:Dongwoo.kang|Dongwoo.kang]]</div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-26712-2672012-12-17T07:55:21Z<p>Dongwoo.kang: </p>
<hr />
<div>__NOEDITSECTION__<br />
__NOTOC__<br />
{{12-267/Navigation}}<br />
==Advanced Ordinary Differential Equations==<br />
===Department of Mathematics, University of Toronto, Fall 2012===<br />
<br />
{{12-267/Crucial Information}}<br />
<br />
===Text===<br />
Boyce and DiPrima, [http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002451.html Elementary Differential Equations and Boundary Value Problems] (current edition is 9th and 10th will be coming out shortly. Hopefully any late enough edition will do).<br />
<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
<br />
* Vitali Kapovitch's 2007 classes: [http://www.math.toronto.edu/vtk/267/ Spring], [http://www.math.toronto.edu/vtk/267fall07/ Fall].<br />
<br />
* Also previously taught by T. Bloom, C. Pugh, D. Remenik.<br />
<br />
* My {{Pensieve Link|Classes/12-267/|12-267 notebook}}.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[[User:Drorbn|Drorbn]] 06:36, 12 September 2012 (EDT): Material by [[User:Syjytg|Syjytg]] moved to [[12-267/Tuesday September 11 Notes]].<br />
<br />
[http://imgur.com/a/OSx1U#0 Summary of techniques to solve differential equations ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Existence_And_Uniqueness_Theorem Fundamental Theorem and Proof from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Derivation_of_Euler-Lagrange Derivation of Euler-Lagrange from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://graphics.ethz.ch/teaching/former/vc_master_06/Downloads/viscomp-varcalc_6.pdf Useful PDF: proof of Euler-Lagrange equation, explanation, examples ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
12-267 [http://math.hunter.cuny.edu/mbenders/cofv.pdf In-depth coverage of Calculus of Variations][[User:Simon1|Simon1]]<br />
<br />
12-267 [http://highered.mcgraw-hill.com/sites/dl/free/007063419x/392340/Calculus_of_Variations.pdf A good summary of Calculus of Variations]<br />
[[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/1vy11#0 All class notes from September 10th to October 5th] [[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/uLSlM Summary of Numerical Methods] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/index.php?title=Numerical_Methods Numerical Methods (wiki)] [[User:Twine|Twine]]<br />
<br />
12-267 [http://imgur.com/a/tliYg Summary of Chapter 3 from the Textbook on Constant Coefficient Second Order ODEs] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/images/e/e6/Geometric_Interpretation_of_Lagrange_Multiplier.pdf Geometric Interpretation of Lagrange Multiplier] [[User:Mathstudent|Mathstudent]]<br />
<br />
Past exams from [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2009] and [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2010] [[User:Twine|Twine]]<br />
<br />
Handwritten notes by [[User:Ktnd3|Ktnd3]]:<br />
<br />
* September: [[Media:Mat267_-_lecture_1(sep.10).PDF|10th]], [[Media:Mat267_-_lecture_2%28sep.11%29.PDF|11th]], [[Media:Mat267_-_lecture_3%28sep.14%29.PDF|14th]], [[Media:Mat267_-_lecture_4%28sep.17%29.PDF|17th]], [[Media:12-267%28lecture5%29.PDF|18th]], [[Media:12-267%28lecture6%29.PDF|21st]], [[Media:12-267%28lecture7%29.PDF|24th]], [[Media:12-267%28lecture8%29.PDF|25th]], [[Media:12-267%28lecture9%29.PDF|28th]]<br />
<br />
* October: [[Media:12-267%28lecture10%29.PDF|1st]], [[Media:12-267%28lecture11%29.PDF|2nd]], [[Media:12-267%28lecture12%29.PDF|5th]], [[Media:12-267%28lecture13%29.PDF|9th]] [[Media:12-267%28lecture14%29.PDF|12th]]<br />
<br />
[http://i.imgur.com/uTugV.jpg Quick guide: system of 1st order linear equations] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Help of Inverse Matrix [http://mathworld.wolfram.com/MatrixInverse.html Matrix Inverse] [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<br />
12-267 [http://imgur.com/a/50sRR All class notes from October 5th to October 30th] [[User:Simon1|Simon1]]<br />
<br />
[http://imgur.com/a/sZSYx#0 Quick guide: Power Series + ODE] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[https://lh4.googleusercontent.com/-V1WnvvzXRlE/TxTGwl_BBjI/AAAAAAAAE9k/yVuy1zRmnWs/Gatos-cute-cute-32-800x500.jpg Mirrors do actually flip top-to-bottom, depending on how you look at them] [[User:jonathanrlove|jonathanrlove]]<br />
<br />
[http://www.guardian.co.uk/notesandqueries/query/0,5753,-19877,00.html Actually, mirrors don't flip top-to-bottom *or* left-to-right] [[User:Twine|Twine]]<br />
<br />
[[12-267/Topic_List|Topics covered this semester]] - [[User:Twine|Twine]]<br />
<br />
(Summary of ODE) Hope it is Helpful.<br />
[[12-267(ODE-Summary).pdf]] - [[User:Dongwoo.kang|Dongwoo.kang]]</div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(ODE-Summary).pdfFile:12-267(ODE-Summary).pdf2012-12-17T07:51:32Z<p>Dongwoo.kang: </p>
<hr />
<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_512-267/Homework Assignment 52012-12-14T02:05:17Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
<br />
This assignment is due in class on Friday November 2. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Consider the following systems of equations:<br />
<br />
<math>A:\begin{cases}<br />
\dot{x}=x-2y & x(0)=3\\<br />
\dot{y}=4y-2x & y(0)=1\\<br />
\end{cases}</math><br />
<br />
<math>B:\begin{cases}<br />
\dot{x}=x-5y & x(0)=3\\<br />
\dot{y}=2x-5y & y(0)=1<br />
\end{cases}</math><br />
<br />
<math>C:\begin{cases}<br />
\dot{x}=y & x(0)=1 \\<br />
\dot{y}=z & y(0)=2 \\<br />
\dot{z}=-6x-11y-6z & z(0)=-1<br />
\end{cases}</math><br />
<br />
# Write each one in a matrix form.<br />
# Find the eigenvalues and eigenvectors of the resulting matrices.<br />
# Diagonalize these matrices.<br />
# Compute <math>e^{tA}</math> for each of those matrices.<br />
# Solve these equations.<br />
<br />
'''Task 2.'''<br />
# Prove that if two matrices <math>A</math> and <math>B</math> satisfy <math>AB=BA</math>, then <math>e^{A+B}=e^Ae^B</math>.<br />
# Find an example for two matrices <math>A</math> and <math>B</math> for which <math>e^{A+B}\neq e^Ae^B</math>.<br />
<br />
'''Task 3.''' Let <math>D</math> be the differential operator <math>\frac{d}{dx}</math>, and let <math>f</math> be a function of the variable <math>x</math> whose Taylor series is convergent everywhere. Write a simple formula for <math>(e^Df)(x)</math>.<br />
<br />
(Here, of course, <math>e^D:=\sum_{k=0}^\infty \frac{D^k}{k!}</math>).<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/txF9Z#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
<br />
Solutions to HW5: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:12-267(HW5-1).jpg|page 1<br />
Image:12-267(HW5-2).jpg|page 2<br />
Image:12-267(HW5-3).jpg|page 3<br />
Image:12-267(HW5-4.jpg|page 4<br />
Image:12-267(HW5-5.jpg|page 5<br />
Image:12-267(HW5-6.jpg|page 6<br />
Image:12-267(HW5-7.jpg|page 7<br />
Image:12-267(HW5-8.jpg|page 8<br />
Image:12-267(HW5-9.jpg|page 9<br />
Image:12-267(HW5-10.jpg|page 10<br />
Image:12-267(HW5-11.jpg|page 11<br />
Image:12-267(HW5-12.jpg|page 12<br />
Image:12-267(HW5-13.jpg|page 13<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-13.jpgFile:12-267(HW5-13.jpg2012-12-14T02:02:57Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-12.jpgFile:12-267(HW5-12.jpg2012-12-14T02:02:50Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-11.jpgFile:12-267(HW5-11.jpg2012-12-14T02:02:43Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-10.jpgFile:12-267(HW5-10.jpg2012-12-14T02:02:25Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-9.jpgFile:12-267(HW5-9.jpg2012-12-14T02:02:06Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-8.jpgFile:12-267(HW5-8.jpg2012-12-14T02:01:57Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-7.jpgFile:12-267(HW5-7.jpg2012-12-14T02:01:47Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-6.jpgFile:12-267(HW5-6.jpg2012-12-14T02:01:38Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-5.jpgFile:12-267(HW5-5.jpg2012-12-14T02:01:31Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-4.jpgFile:12-267(HW5-4.jpg2012-12-14T02:01:23Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-3).jpgFile:12-267(HW5-3).jpg2012-12-14T02:01:15Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-2).jpgFile:12-267(HW5-2).jpg2012-12-14T02:01:07Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW5-1).jpgFile:12-267(HW5-1).jpg2012-12-14T02:00:57Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/The_Final_Exam12-267/The Final Exam2012-12-13T21:40:54Z<p>Dongwoo.kang: </p>
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<div>{{12-267/Navigation}}<br />
<br />
Our final exam is coming up. It will take place on Tuesday December 18th, from 7PM until 10PM (believe me that wasn't my decision), at East Hall, University College, 15 King's College Circle. <br />
<br />
===Sample Final===<br />
See {{pensieve link|Classes/12-267/SampleFinal.pdf|SampleFinal.pdf}}.<br />
<br />
===Content and Style===<br />
It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:<br />
*Basic techniques: linear, separated, exact equations, integrating factors, etc.<br />
*The Fundamental Theorem (existence and uniqueness).<br />
*Calculus of Variations.<br />
*Numerical methods.<br />
*High order linear equations with constant coefficients, "undetermined coefficients".<br />
*Systems of linear equations with constant coefficients, matrix exponentiation, phase portraits, Wronskians, non-homogeneous systems.<br />
*Series solutions and regular singular points.<br />
*Qualitative analysis.<br />
<br />
As for the style -<br />
<br />
*You can expect to be asked to reproduce some proofs that were given in class.<br />
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.<br />
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.<br />
*You can expect some calculations (but nothing that will require a calculator).<br />
<br />
Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.<br />
<br />
'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>x\in[0,1]</math>", does it mean "choose <math>x\in[0,1]</math>", or "we've just proven that <math>x\in[0,1]</math>", or "assume by contradiction that <math>x\in[0,1]</math>", or "for every <math>x\in[0,1]</math>" or "there exists <math>x\in[0,1]</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!<br />
<br />
==Office Hours==<br />
Jordan and I will hold pre-exam office hours as follows:<br />
* Friday December 14, 3PM-5:30PM, with Jordan at the Math Lounge (the round room on the 6th floor of Bahen).<br />
* Monday December 17, 10:30AM-11:30AM, with Dror at Bahen 6178.<br />
* Monday December 17, 3PM-5:30PM, with Jordan at the Math Lounge.<br />
* Tuesday December 18, 10AM-Noon and 3PM-5PM, with Dror at Bahen 6178.</div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_812-267/Homework Assignment 82012-12-10T03:55:24Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
This assignment is due in class on Tuesday November 27. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Find the singular (that is, non-ordinary) points of the equation below, and for each one decide if it is regular or not:<br />
<center><math>x(1-x^2)^3y''+(1-x^2)^2y'+2(1+x)y=0</math></center><br />
<br />
'''Task 2.''' Find the general solution of the following two equations:<br />
# <math>x^2y''-3xy'+4y=0</math><br />
# <math>x^2y''+2xy'+y=0</math><br />
(You are allowed to use complex numbers within the derivation, but your solutions should be real-valued).<br />
<br />
'''Task 3.''' Using power series, find <u>two</u> linearly independent solutions for each of the equations<br />
# <math>2xy''+y'+xy=0</math><br />
# <math>x^2y''+xy'+2xy=0</math><br />
<br />
'''Task 4.''' An equation <math>p(x)y''+q(x)y'+r(x)y=0</math> is said to have a regular singular at <math>x=\infty</math> if the equation obtained from it by the change of substitution <math>x=1/t</math> has a regular singular point at <math>t=0</math>. Write explicitly the conditions on <math>p</math>, <math>q</math>, and <math>r</math> that this entails.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/xEiAw#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Solutions to HW8: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:12-267(HW8-1).jpg|page 1<br />
Image:12-267(HW8-2).jpg|page 2<br />
Image:12-267(HW8-3).jpg|page 3<br />
Image:12-267(HW8-4).jpg|page 4<br />
Image:12-267(HW8-5).jpg|page 5<br />
Image:12-267(HW8-6).jpg|page 6<br />
Image:12-267(HW8-7).jpg|page 7<br />
Image:12-267(HW8-8).jpg|page 8<br />
Image:12-267(HW8-9).jpg|page 9<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_612-267/Homework Assignment 62012-12-10T03:53:59Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
<br />
This assignment is due in class on Friday November 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Draw the phase portraits for the following systems, near <math>(x,y)=(0,0)</math>:<br />
# <math>\begin{cases} \dot{x}=2x+y \\ \dot{y}=-x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=4x-5y \\ \dot{y}=4x-4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=x-2y \\ \dot{y}=-2x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-x+y \\ \dot{y}=-5x+3y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-5x+4y \\ \dot{y}=-8x+7y \end{cases}</math>.<br />
<br />
'''Task 2.''' Draw the phase portrait of the system<br />
<center><br />
<math>\begin{cases}\dot{x}=17+x-9y+\sin(2-2x-y+xy)\\\dot{y}=7+2x-5y+\cos(x-1)\end{cases}</math><br />
</center><br />
near the point <math>(x,y)=(1,2)</math>.<br />
<br />
'''Task 3.''' Solve using diagonalization (one solution is enough):<br />
# <math>v'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} e^t \\ t \end{pmatrix}</math>.<br />
# <math>v'=\begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}v + \begin{pmatrix} -\cos t \\ \sin t \end{pmatrix}</math>.<br />
<br />
'''Task 4.''' Assume <math>t>0</math>. For the following equation,<br />
<center><math>tv'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} 1-t^2 \\ 2t \end{pmatrix}</math></center>,<br />
it is given that a solution of the homogeneous version is<br />
<center><math>v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t + \begin{pmatrix}1\\3\end{pmatrix}t^{-1}</math>.</center><br />
Use "fundamental solutions" to find a solution of the full equation.<br />
<br />
'''Task 5.''' (Not for grade). Find a quadratic differential equation whose phase portrait is as below.<br />
<br />
[[Image:12-267-MonkeySaddleFlow.png|center|400px]]<br />
<br />
'''Hint.''' "Monkey Saddle".<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/53nSl#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Solutions to HW6: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:12-267(HW6-1).jpg|page 1<br />
Image:12-267(HW6-2).jpg|page 2<br />
Image:12-267(HW6-3).jpg|page 3<br />
Image:12-267(HW6-4).jpg|page 4<br />
Image:12-267(HW6-5).jpg|page 5<br />
Image:12-267(HW6-6).jpg|page 6<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-9).jpgFile:12-267(HW8-9).jpg2012-12-10T03:52:18Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-8).jpgFile:12-267(HW8-8).jpg2012-12-10T03:52:11Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-7).jpgFile:12-267(HW8-7).jpg2012-12-10T03:52:04Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-6).jpgFile:12-267(HW8-6).jpg2012-12-10T03:51:56Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-5).jpgFile:12-267(HW8-5).jpg2012-12-10T03:51:50Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-4).jpgFile:12-267(HW8-4).jpg2012-12-10T03:51:17Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-3).jpgFile:12-267(HW8-3).jpg2012-12-10T03:51:08Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-2).jpgFile:12-267(HW8-2).jpg2012-12-10T03:51:01Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW8-1).jpgFile:12-267(HW8-1).jpg2012-12-10T03:50:48Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW6-6).jpgFile:12-267(HW6-6).jpg2012-12-10T03:50:23Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW6-5).jpgFile:12-267(HW6-5).jpg2012-12-10T03:50:15Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW6-4).jpgFile:12-267(HW6-4).jpg2012-12-10T03:50:07Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW6-3).jpgFile:12-267(HW6-3).jpg2012-12-10T03:49:58Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW6-2).jpgFile:12-267(HW6-2).jpg2012-12-10T03:49:50Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=File:12-267(HW6-1).jpgFile:12-267(HW6-1).jpg2012-12-10T03:49:37Z<p>Dongwoo.kang: </p>
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<div></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_812-267/Homework Assignment 82012-12-07T14:52:45Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
This assignment is due in class on Tuesday November 27. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Find the singular (that is, non-ordinary) points of the equation below, and for each one decide if it is regular or not:<br />
<center><math>x(1-x^2)^3y''+(1-x^2)^2y'+2(1+x)y=0</math></center><br />
<br />
'''Task 2.''' Find the general solution of the following two equations:<br />
# <math>x^2y''-3xy'+4y=0</math><br />
# <math>x^2y''+2xy'+y=0</math><br />
(You are allowed to use complex numbers within the derivation, but your solutions should be real-valued).<br />
<br />
'''Task 3.''' Using power series, find <u>two</u> linearly independent solutions for each of the equations<br />
# <math>2xy''+y'+xy=0</math><br />
# <math>x^2y''+xy'+2xy=0</math><br />
<br />
'''Task 4.''' An equation <math>p(x)y''+q(x)y'+r(x)y=0</math> is said to have a regular singular at <math>x=\infty</math> if the equation obtained from it by the change of substitution <math>x=1/t</math> has a regular singular point at <math>t=0</math>. Write explicitly the conditions on <math>p</math>, <math>q</math>, and <math>r</math> that this entails.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/xEiAw#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Solutions to HW8: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:Hw08-1.jpg|page 1<br />
Image:Hw08-2.jpg|page 2<br />
Image:Hw08-3.jpg|page 3<br />
Image:Hw08-4.jpg|page 4<br />
Image:Hw08-5.jpg|page 5<br />
Image:Hw08-6.jpg|page 6<br />
Image:Hw08-7.jpg|page 7<br />
Image:Hw08-8.jpg|page 8<br />
Image:Hw08-9.jpg|page 9<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_612-267/Homework Assignment 62012-12-06T22:19:58Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
<br />
This assignment is due in class on Friday November 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Draw the phase portraits for the following systems, near <math>(x,y)=(0,0)</math>:<br />
# <math>\begin{cases} \dot{x}=2x+y \\ \dot{y}=-x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=4x-5y \\ \dot{y}=4x-4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=x-2y \\ \dot{y}=-2x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-x+y \\ \dot{y}=-5x+3y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-5x+4y \\ \dot{y}=-8x+7y \end{cases}</math>.<br />
<br />
'''Task 2.''' Draw the phase portrait of the system<br />
<center><br />
<math>\begin{cases}\dot{x}=17+x-9y+\sin(2-2x-y+xy)\\\dot{y}=7+2x-5y+\cos(x-1)\end{cases}</math><br />
</center><br />
near the point <math>(x,y)=(1,2)</math>.<br />
<br />
'''Task 3.''' Solve using diagonalization (one solution is enough):<br />
# <math>v'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} e^t \\ t \end{pmatrix}</math>.<br />
# <math>v'=\begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}v + \begin{pmatrix} -\cos t \\ \sin t \end{pmatrix}</math>.<br />
<br />
'''Task 4.''' Assume <math>t>0</math>. For the following equation,<br />
<center><math>tv'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} 1-t^2 \\ 2t \end{pmatrix}</math></center>,<br />
it is given that a solution of the homogeneous version is<br />
<center><math>v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t + \begin{pmatrix}1\\3\end{pmatrix}t^{-1}</math>.</center><br />
Use "fundamental solutions" to find a solution of the full equation.<br />
<br />
'''Task 5.''' (Not for grade). Find a quadratic differential equation whose phase portrait is as below.<br />
<br />
[[Image:12-267-MonkeySaddleFlow.png|center|400px]]<br />
<br />
'''Hint.''' "Monkey Saddle".<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/53nSl#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Solutions to HW6: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:hw6-1.jpg|page 1<br />
Image:hw6-2.jpg|page 2<br />
Image:hw6-3.jpg|page 3<br />
Image:hw6-4.jpg|page 4<br />
Image:hw6-5.jpg|page 5<br />
Image:hw6-6.jpg|page 6<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_812-267/Homework Assignment 82012-12-06T22:18:17Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
This assignment is due in class on Tuesday November 27. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Find the singular (that is, non-ordinary) points of the equation below, and for each one decide if it is regular or not:<br />
<center><math>x(1-x^2)^3y''+(1-x^2)^2y'+2(1+x)y=0</math></center><br />
<br />
'''Task 2.''' Find the general solution of the following two equations:<br />
# <math>x^2y''-3xy'+4y=0</math><br />
# <math>x^2y''+2xy'+y=0</math><br />
(You are allowed to use complex numbers within the derivation, but your solutions should be real-valued).<br />
<br />
'''Task 3.''' Using power series, find <u>two</u> linearly independent solutions for each of the equations<br />
# <math>2xy''+y'+xy=0</math><br />
# <math>x^2y''+xy'+2xy=0</math><br />
<br />
'''Task 4.''' An equation <math>p(x)y''+q(x)y'+r(x)y=0</math> is said to have a regular singular at <math>x=\infty</math> if the equation obtained from it by the change of substitution <math>x=1/t</math> has a regular singular point at <math>t=0</math>. Write explicitly the conditions on <math>p</math>, <math>q</math>, and <math>r</math> that this entails.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/xEiAw#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Solutions to HW8: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:hw08-1.jpg|page 1<br />
Image:hw08-2.jpg|page 2<br />
Image:hw08-3.jpg|page 3<br />
Image:hw08-4.jpg|page 4<br />
Image:hw08-5.jpg|page 5<br />
Image:hw08-6.jpg|page 6<br />
Image:hw08-7.jpg|page 7<br />
Image:hw08-8.jpg|page 8<br />
Image:hw08-9.jpg|page 9<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_612-267/Homework Assignment 62012-12-06T22:17:55Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
<br />
This assignment is due in class on Friday November 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Draw the phase portraits for the following systems, near <math>(x,y)=(0,0)</math>:<br />
# <math>\begin{cases} \dot{x}=2x+y \\ \dot{y}=-x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=4x-5y \\ \dot{y}=4x-4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=x-2y \\ \dot{y}=-2x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-x+y \\ \dot{y}=-5x+3y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-5x+4y \\ \dot{y}=-8x+7y \end{cases}</math>.<br />
<br />
'''Task 2.''' Draw the phase portrait of the system<br />
<center><br />
<math>\begin{cases}\dot{x}=17+x-9y+\sin(2-2x-y+xy)\\\dot{y}=7+2x-5y+\cos(x-1)\end{cases}</math><br />
</center><br />
near the point <math>(x,y)=(1,2)</math>.<br />
<br />
'''Task 3.''' Solve using diagonalization (one solution is enough):<br />
# <math>v'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} e^t \\ t \end{pmatrix}</math>.<br />
# <math>v'=\begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}v + \begin{pmatrix} -\cos t \\ \sin t \end{pmatrix}</math>.<br />
<br />
'''Task 4.''' Assume <math>t>0</math>. For the following equation,<br />
<center><math>tv'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} 1-t^2 \\ 2t \end{pmatrix}</math></center>,<br />
it is given that a solution of the homogeneous version is<br />
<center><math>v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t + \begin{pmatrix}1\\3\end{pmatrix}t^{-1}</math>.</center><br />
Use "fundamental solutions" to find a solution of the full equation.<br />
<br />
'''Task 5.''' (Not for grade). Find a quadratic differential equation whose phase portrait is as below.<br />
<br />
[[Image:12-267-MonkeySaddleFlow.png|center|400px]]<br />
<br />
'''Hint.''' "Monkey Saddle".<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/53nSl#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]</div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_612-267/Homework Assignment 62012-12-06T22:16:52Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
<br />
This assignment is due in class on Friday November 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Draw the phase portraits for the following systems, near <math>(x,y)=(0,0)</math>:<br />
# <math>\begin{cases} \dot{x}=2x+y \\ \dot{y}=-x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=4x-5y \\ \dot{y}=4x-4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=x-2y \\ \dot{y}=-2x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-x+y \\ \dot{y}=-5x+3y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-5x+4y \\ \dot{y}=-8x+7y \end{cases}</math>.<br />
<br />
'''Task 2.''' Draw the phase portrait of the system<br />
<center><br />
<math>\begin{cases}\dot{x}=17+x-9y+\sin(2-2x-y+xy)\\\dot{y}=7+2x-5y+\cos(x-1)\end{cases}</math><br />
</center><br />
near the point <math>(x,y)=(1,2)</math>.<br />
<br />
'''Task 3.''' Solve using diagonalization (one solution is enough):<br />
# <math>v'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} e^t \\ t \end{pmatrix}</math>.<br />
# <math>v'=\begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}v + \begin{pmatrix} -\cos t \\ \sin t \end{pmatrix}</math>.<br />
<br />
'''Task 4.''' Assume <math>t>0</math>. For the following equation,<br />
<center><math>tv'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} 1-t^2 \\ 2t \end{pmatrix}</math></center>,<br />
it is given that a solution of the homogeneous version is<br />
<center><math>v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t + \begin{pmatrix}1\\3\end{pmatrix}t^{-1}</math>.</center><br />
Use "fundamental solutions" to find a solution of the full equation.<br />
<br />
'''Task 5.''' (Not for grade). Find a quadratic differential equation whose phase portrait is as below.<br />
<br />
[[Image:12-267-MonkeySaddleFlow.png|center|400px]]<br />
<br />
'''Hint.''' "Monkey Saddle".<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/53nSl#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Solutions to HW8: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:hw08-1.jpg|page 1<br />
Image:hw08-2.jpg|page 2<br />
Image:hw08-3.jpg|page 3<br />
Image:hw08-4.jpg|page 4<br />
Image:hw08-5.jpg|page 5<br />
Image:hw08-6.jpg|page 6<br />
Image:hw08-7.jpg|page 7<br />
Image:hw08-8.jpg|page 8<br />
Image:hw08-9.jpg|page 9<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_612-267/Homework Assignment 62012-12-06T22:13:12Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
<br />
This assignment is due in class on Friday November 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Draw the phase portraits for the following systems, near <math>(x,y)=(0,0)</math>:<br />
# <math>\begin{cases} \dot{x}=2x+y \\ \dot{y}=-x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=4x-5y \\ \dot{y}=4x-4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=x-2y \\ \dot{y}=-2x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-x+y \\ \dot{y}=-5x+3y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-5x+4y \\ \dot{y}=-8x+7y \end{cases}</math>.<br />
<br />
'''Task 2.''' Draw the phase portrait of the system<br />
<center><br />
<math>\begin{cases}\dot{x}=17+x-9y+\sin(2-2x-y+xy)\\\dot{y}=7+2x-5y+\cos(x-1)\end{cases}</math><br />
</center><br />
near the point <math>(x,y)=(1,2)</math>.<br />
<br />
'''Task 3.''' Solve using diagonalization (one solution is enough):<br />
# <math>v'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} e^t \\ t \end{pmatrix}</math>.<br />
# <math>v'=\begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}v + \begin{pmatrix} -\cos t \\ \sin t \end{pmatrix}</math>.<br />
<br />
'''Task 4.''' Assume <math>t>0</math>. For the following equation,<br />
<center><math>tv'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} 1-t^2 \\ 2t \end{pmatrix}</math></center>,<br />
it is given that a solution of the homogeneous version is<br />
<center><math>v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t + \begin{pmatrix}1\\3\end{pmatrix}t^{-1}</math>.</center><br />
Use "fundamental solutions" to find a solution of the full equation.<br />
<br />
'''Task 5.''' (Not for grade). Find a quadratic differential equation whose phase portrait is as below.<br />
<br />
[[Image:12-267-MonkeySaddleFlow.png|center|400px]]<br />
<br />
'''Hint.''' "Monkey Saddle".<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/53nSl#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Solutions to HW6: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:Hw6-1.jpg|page 1<br />
Image:Hw6-2.jpg|page 2<br />
Image:Hw6-3.jpg|page 3<br />
Image:Hw6-4.jpg|page 4<br />
Image:Hw6-5.jpg|page 5<br />
Image:Hw6-6.jpg|page 6<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Homework_Assignment_612-267/Homework Assignment 62012-12-06T22:12:02Z<p>Dongwoo.kang: </p>
<hr />
<div>{{12-267/Navigation}}<br />
<br />
This assignment is due in class on Friday November 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.<br />
<br />
'''Task 1.''' Draw the phase portraits for the following systems, near <math>(x,y)=(0,0)</math>:<br />
# <math>\begin{cases} \dot{x}=2x+y \\ \dot{y}=-x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=4x-5y \\ \dot{y}=4x-4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=x-2y \\ \dot{y}=-2x+4y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-x+y \\ \dot{y}=-5x+3y \end{cases}</math>.<br />
# <math>\begin{cases} \dot{x}=-5x+4y \\ \dot{y}=-8x+7y \end{cases}</math>.<br />
<br />
'''Task 2.''' Draw the phase portrait of the system<br />
<center><br />
<math>\begin{cases}\dot{x}=17+x-9y+\sin(2-2x-y+xy)\\\dot{y}=7+2x-5y+\cos(x-1)\end{cases}</math><br />
</center><br />
near the point <math>(x,y)=(1,2)</math>.<br />
<br />
'''Task 3.''' Solve using diagonalization (one solution is enough):<br />
# <math>v'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} e^t \\ t \end{pmatrix}</math>.<br />
# <math>v'=\begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}v + \begin{pmatrix} -\cos t \\ \sin t \end{pmatrix}</math>.<br />
<br />
'''Task 4.''' Assume <math>t>0</math>. For the following equation,<br />
<center><math>tv'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} 1-t^2 \\ 2t \end{pmatrix}</math></center>,<br />
it is given that a solution of the homogeneous version is<br />
<center><math>v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t + \begin{pmatrix}1\\3\end{pmatrix}t^{-1}</math>.</center><br />
Use "fundamental solutions" to find a solution of the full equation.<br />
<br />
'''Task 5.''' (Not for grade). Find a quadratic differential equation whose phase portrait is as below.<br />
<br />
[[Image:12-267-MonkeySaddleFlow.png|center|400px]]<br />
<br />
'''Hint.''' "Monkey Saddle".<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[http://imgur.com/a/53nSl#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Solutions to HW1: [[User:Dongwoo.kang|Dongwoo.kang]]<br />
<gallery><br />
Image:Hw6-1.jpg|page 1<br />
Image:Hw6-2.jpg|page 2<br />
Image:Hw6-3.jpg|page 3<br />
Image:Hw6-4.jpg|page 4<br />
Image:Hw6-5.jpg|page 5<br />
Image:Hw6-6.jpg|page 6<br />
</gallery></div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-26712-2672012-11-03T00:21:05Z<p>Dongwoo.kang: </p>
<hr />
<div>__NOEDITSECTION__<br />
__NOTOC__<br />
{{12-267/Navigation}}<br />
==Advanced Ordinary Differential Equations==<br />
===Department of Mathematics, University of Toronto, Fall 2012===<br />
<br />
{{12-267/Crucial Information}}<br />
<br />
===Text===<br />
Boyce and DiPrima, [http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002451.html Elementary Differential Equations and Boundary Value Problems] (current edition is 9th and 10th will be coming out shortly. Hopefully any late enough edition will do).<br />
<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
<br />
* Vitali Kapovitch's 2007 classes: [http://www.math.toronto.edu/vtk/267/ Spring], [http://www.math.toronto.edu/vtk/267fall07/ Fall].<br />
<br />
* Also previously taught by T. Bloom, C. Pugh, D. Remenik.<br />
<br />
* My {{Pensieve Link|Classes/12-267/|12-267 notebook}}.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[[User:Drorbn|Drorbn]] 06:36, 12 September 2012 (EDT): Material by [[User:Syjytg|Syjytg]] moved to [[12-267/Tuesday September 11 Notes]].<br />
<br />
[http://imgur.com/a/OSx1U#0 Summary of techniques to solve differential equations ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Existence_And_Uniqueness_Theorem Fundamental Theorem and Proof from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Derivation_of_Euler-Lagrange Derivation of Euler-Lagrange from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://graphics.ethz.ch/teaching/former/vc_master_06/Downloads/viscomp-varcalc_6.pdf Useful PDF: proof of Euler-Lagrange equation, explanation, examples ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
12-267 [http://math.hunter.cuny.edu/mbenders/cofv.pdf In-depth coverage of Calculus of Variations][[User:Simon1|Simon1]]<br />
<br />
12-267 [http://highered.mcgraw-hill.com/sites/dl/free/007063419x/392340/Calculus_of_Variations.pdf A good summary of Calculus of Variations]<br />
[[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/1vy11#0 All class notes from September 10th to October 5th] [[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/uLSlM Summary of Numerical Methods] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/index.php?title=Numerical_Methods Numerical Methods (wiki)] [[User:Twine|Twine]]<br />
<br />
12-267 [http://imgur.com/a/tliYg Summary of Chapter 3 from the Textbook on Constant Coefficient Second Order ODEs] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/images/e/e6/Geometric_Interpretation_of_Lagrange_Multiplier.pdf Geometric Interpretation of Lagrange Multiplier] [[User:Mathstudent|Mathstudent]]<br />
<br />
Past exams from [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2009] and [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2010] [[User:Twine|Twine]]<br />
<br />
Handwritten notes by [[User:Ktnd3|Ktnd3]]:<br />
<br />
* September: [[Media:Mat267_-_lecture_1(sep.10).PDF|10th]], [[Media:Mat267_-_lecture_2%28sep.11%29.PDF|11th]], [[Media:Mat267_-_lecture_3%28sep.14%29.PDF|14th]], [[Media:Mat267_-_lecture_4%28sep.17%29.PDF|17th]], [[Media:12-267%28lecture5%29.PDF|18th]], [[Media:12-267%28lecture6%29.PDF|21st]], [[Media:12-267%28lecture7%29.PDF|24th]], [[Media:12-267%28lecture8%29.PDF|25th]], [[Media:12-267%28lecture9%29.PDF|28th]]<br />
<br />
* October: [[Media:12-267%28lecture10%29.PDF|1st]], [[Media:12-267%28lecture11%29.PDF|2nd]], [[Media:12-267%28lecture12%29.PDF|5th]], [[Media:12-267%28lecture13%29.PDF|9th]] [[Media:12-267%28lecture14%29.PDF|12th]]<br />
<br />
[http://i.imgur.com/uTugV.jpg Quick guide: system of 1st order linear equations] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Help of Inverse Matrix [http://mathworld.wolfram.com/MatrixInverse.html Matrix Inverse] [[User:Dongwoo.kang|Dongwoo.kang]]</div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-26712-2672012-11-03T00:18:59Z<p>Dongwoo.kang: </p>
<hr />
<div>__NOEDITSECTION__<br />
__NOTOC__<br />
{{12-267/Navigation}}<br />
==Advanced Ordinary Differential Equations==<br />
===Department of Mathematics, University of Toronto, Fall 2012===<br />
<br />
{{12-267/Crucial Information}}<br />
<br />
===Text===<br />
Boyce and DiPrima, [http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002451.html Elementary Differential Equations and Boundary Value Problems] (current edition is 9th and 10th will be coming out shortly. Hopefully any late enough edition will do).<br />
<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
<br />
* Vitali Kapovitch's 2007 classes: [http://www.math.toronto.edu/vtk/267/ Spring], [http://www.math.toronto.edu/vtk/267fall07/ Fall].<br />
<br />
* Also previously taught by T. Bloom, C. Pugh, D. Remenik.<br />
<br />
* My {{Pensieve Link|Classes/12-267/|12-267 notebook}}.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[[User:Drorbn|Drorbn]] 06:36, 12 September 2012 (EDT): Material by [[User:Syjytg|Syjytg]] moved to [[12-267/Tuesday September 11 Notes]].<br />
<br />
[http://imgur.com/a/OSx1U#0 Summary of techniques to solve differential equations ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Existence_And_Uniqueness_Theorem Fundamental Theorem and Proof from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Derivation_of_Euler-Lagrange Derivation of Euler-Lagrange from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://graphics.ethz.ch/teaching/former/vc_master_06/Downloads/viscomp-varcalc_6.pdf Useful PDF: proof of Euler-Lagrange equation, explanation, examples ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
12-267 [http://math.hunter.cuny.edu/mbenders/cofv.pdf In-depth coverage of Calculus of Variations][[User:Simon1|Simon1]]<br />
<br />
12-267 [http://highered.mcgraw-hill.com/sites/dl/free/007063419x/392340/Calculus_of_Variations.pdf A good summary of Calculus of Variations]<br />
[[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/1vy11#0 All class notes from September 10th to October 5th] [[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/uLSlM Summary of Numerical Methods] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/index.php?title=Numerical_Methods Numerical Methods (wiki)] [[User:Twine|Twine]]<br />
<br />
12-267 [http://imgur.com/a/tliYg Summary of Chapter 3 from the Textbook on Constant Coefficient Second Order ODEs] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/images/e/e6/Geometric_Interpretation_of_Lagrange_Multiplier.pdf Geometric Interpretation of Lagrange Multiplier] [[User:Mathstudent|Mathstudent]]<br />
<br />
Past exams from [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2009] and [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2010] [[User:Twine|Twine]]<br />
<br />
Handwritten notes by [[User:Ktnd3|Ktnd3]]:<br />
<br />
* September: [[Media:Mat267_-_lecture_1(sep.10).PDF|10th]], [[Media:Mat267_-_lecture_2%28sep.11%29.PDF|11th]], [[Media:Mat267_-_lecture_3%28sep.14%29.PDF|14th]], [[Media:Mat267_-_lecture_4%28sep.17%29.PDF|17th]], [[Media:12-267%28lecture5%29.PDF|18th]], [[Media:12-267%28lecture6%29.PDF|21st]], [[Media:12-267%28lecture7%29.PDF|24th]], [[Media:12-267%28lecture8%29.PDF|25th]], [[Media:12-267%28lecture9%29.PDF|28th]]<br />
<br />
* October: [[Media:12-267%28lecture10%29.PDF|1st]], [[Media:12-267%28lecture11%29.PDF|2nd]], [[Media:12-267%28lecture12%29.PDF|5th]], [[Media:12-267%28lecture13%29.PDF|9th]] [[Media:12-267%28lecture14%29.PDF|12th]]<br />
<br />
[http://i.imgur.com/uTugV.jpg Quick guide: system of 1st order linear equations] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
Help of Inverse Matrix [http://mathworld.wolfram.com/MatrixInverse.html Inverse Matrix] [[User:Dongwoo.kang|Dongwoo.kang]]</div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-26712-2672012-11-03T00:17:42Z<p>Dongwoo.kang: </p>
<hr />
<div>__NOEDITSECTION__<br />
__NOTOC__<br />
{{12-267/Navigation}}<br />
==Advanced Ordinary Differential Equations==<br />
===Department of Mathematics, University of Toronto, Fall 2012===<br />
<br />
{{12-267/Crucial Information}}<br />
<br />
===Text===<br />
Boyce and DiPrima, [http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002451.html Elementary Differential Equations and Boundary Value Problems] (current edition is 9th and 10th will be coming out shortly. Hopefully any late enough edition will do).<br />
<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
<br />
* Vitali Kapovitch's 2007 classes: [http://www.math.toronto.edu/vtk/267/ Spring], [http://www.math.toronto.edu/vtk/267fall07/ Fall].<br />
<br />
* Also previously taught by T. Bloom, C. Pugh, D. Remenik.<br />
<br />
* My {{Pensieve Link|Classes/12-267/|12-267 notebook}}.<br />
<br />
{{Template:12-267:Dror/Students Divider}}<br />
<br />
[[User:Drorbn|Drorbn]] 06:36, 12 September 2012 (EDT): Material by [[User:Syjytg|Syjytg]] moved to [[12-267/Tuesday September 11 Notes]].<br />
<br />
[http://imgur.com/a/OSx1U#0 Summary of techniques to solve differential equations ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Existence_And_Uniqueness_Theorem Fundamental Theorem and Proof from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://drorbn.net/index.php?title=12-267/Derivation_of_Euler-Lagrange Derivation of Euler-Lagrange from Lecture] [[User:Twine|Twine]]<br />
<br />
[http://graphics.ethz.ch/teaching/former/vc_master_06/Downloads/viscomp-varcalc_6.pdf Useful PDF: proof of Euler-Lagrange equation, explanation, examples ] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
<br />
12-267 [http://math.hunter.cuny.edu/mbenders/cofv.pdf In-depth coverage of Calculus of Variations][[User:Simon1|Simon1]]<br />
<br />
12-267 [http://highered.mcgraw-hill.com/sites/dl/free/007063419x/392340/Calculus_of_Variations.pdf A good summary of Calculus of Variations]<br />
[[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/1vy11#0 All class notes from September 10th to October 5th] [[User:Simon1|Simon1]]<br />
<br />
12-267 [http://imgur.com/a/uLSlM Summary of Numerical Methods] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/index.php?title=Numerical_Methods Numerical Methods (wiki)] [[User:Twine|Twine]]<br />
<br />
12-267 [http://imgur.com/a/tliYg Summary of Chapter 3 from the Textbook on Constant Coefficient Second Order ODEs] [[User:Simon1|Simon1]]<br />
<br />
[http://drorbn.net/images/e/e6/Geometric_Interpretation_of_Lagrange_Multiplier.pdf Geometric Interpretation of Lagrange Multiplier] [[User:Mathstudent|Mathstudent]]<br />
<br />
Past exams from [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2009] and [http://exams.library.utoronto.ca.myaccess.library.utoronto.ca/handle/exams/4429 December 2010] [[User:Twine|Twine]]<br />
<br />
Handwritten notes by [[User:Ktnd3|Ktnd3]]:<br />
<br />
* September: [[Media:Mat267_-_lecture_1(sep.10).PDF|10th]], [[Media:Mat267_-_lecture_2%28sep.11%29.PDF|11th]], [[Media:Mat267_-_lecture_3%28sep.14%29.PDF|14th]], [[Media:Mat267_-_lecture_4%28sep.17%29.PDF|17th]], [[Media:12-267%28lecture5%29.PDF|18th]], [[Media:12-267%28lecture6%29.PDF|21st]], [[Media:12-267%28lecture7%29.PDF|24th]], [[Media:12-267%28lecture8%29.PDF|25th]], [[Media:12-267%28lecture9%29.PDF|28th]]<br />
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* October: [[Media:12-267%28lecture10%29.PDF|1st]], [[Media:12-267%28lecture11%29.PDF|2nd]], [[Media:12-267%28lecture12%29.PDF|5th]], [[Media:12-267%28lecture13%29.PDF|9th]] [[Media:12-267%28lecture14%29.PDF|12th]]<br />
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[http://i.imgur.com/uTugV.jpg Quick guide: system of 1st order linear equations] [[User:Vsbdthrsh|Vsbdthrsh]]<br />
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Help of Inverse Matrix<br />
[http://mathworld.wolfram.com/MatrixInverse.html] [[User:Dongwoo.kang|Dongwoo.kang]]</div>Dongwoo.kanghttp://drorbn.net/index.php?title=12-267/Class_Photo12-267/Class Photo2012-10-02T08:19:02Z<p>Dongwoo.kang: /* Who We Are... */</p>
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<div>Our class on September 25, 2012; photo by Jordan Bell:<br />
[[Image:12-267-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]<br />
{{12-267/Navigation}}<br />
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Please identify yourself in this photo! There are two ways to do that:<br />
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* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.<br />
* Send [[User:Drorbn|Dror]] an email message with this information.<br />
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The first option is more fun but less private.<br />
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===Who We Are...===<br />
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{| align=center border=1 cellspacing=0<br />
|-<br />
!First name<br />
!Last name<br />
!UserID<br />
!Email<br />
!In the photo<br />
!Comments<br />
<br />
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=left most person in the front row, red shirt with red and white stripes|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}<br />
{{Photo Entry|last=Gordon|first=Jake|userid=Gordonj|email=jake. gordon @mail. utoronto. ca|location= second row from bottom, second from the left in the green sweater.|comments=}}<br />
{{Photo Entry|last=Jaffe|first=Ethan|userid=Eyjaffe|email=ethan.jaffe @mail. utoronto.ca|location=top row, eighth from right, directly below the "RIES" in "Laboratories."|comments=}}<br />
{{Photo Entry|last=Kang|first=Dongwoo (Daniel)|userid=Dongwoo.kang|email=dongwoo.kang@ mail.utoronto.ca|location= 2nd from the right, first row, the asian guy with leather jacket and red shirt |comments=}}<br />
{{Photo Entry|last=Kojar|first=Tomas|userid=|email=tomas. kojar @mail. utoronto. ca|location=fourth row, fourth from right, the guy not looking at the camera.|comments=}}<br />
{{Photo Entry|last=Love|first=Jonathan|userid=Jonathanrlove|email=jonathan.love@ mail.utoronto.ca|location=immediately to the left of the big awkward space in the middle row|comments=}}<br />
{{Photo Entry|last=Naranjo|first=Alvaro|userid=naranjoa|email=alvaro. naranjo @ mail.utoronto.ca|location=directly under Y in "RAMSAY" wearing red sweater|comments=}}<br />
{{Photo Entry|last=Wei|first=Mian|userid=Mianwei|email=mian. wei@ mail.utoronto.ca|location=4th from the right, last row, the one with half his face covered |comments=}}<br />
<br />
<!--PLEASE KEEP IN ALPHABETICAL ORDER, BY LAST NAME--><br />
|}<br />
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<!--PLEASE KEEP IN ALPHABETICAL ORDER, BY LAST NAME--></div>Dongwoo.kang