Drorbn - User contributions [en]

12-267/Homework Assignment 8

2012-12-04T21:01:58Z

Vsbdthrsh:

{{12-267/Navigation}}
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.

'''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:
<center><math>x(1-x^2)^3y''+(1-x^2)^2y'+2(1+x)y=0</math></center>

'''Task 2.''' Find the general solution of the following two equations:
# <math>x^2y''-3xy'+4y=0</math>
# <math>x^2y''+2xy'+y=0</math>
(You are allowed to use complex numbers within the derivation, but your solutions should be real-valued).

'''Task 3.''' Using power series, find <u>two</u> linearly independent solutions for each of the equations
# <math>2xy''+y'+xy=0</math>
# <math>x^2y''+xy'+2xy=0</math>

'''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.

{{Template:12-267:Dror/Students Divider}}

[http://imgur.com/a/xEiAw#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]

12-267/Homework Assignment 7

2012-12-01T02:11:20Z

Vsbdthrsh:

{{12-267/Navigation}}
This assignment is due in class on <span style="color: red;">Friday November 23</span>. 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.

'''Task 1.''' Find the radius of convergence of the series
# <math>\sum_{n=0}^\infty 2^nx^n</math>.
# <math>\sum_{n=0}^\infty\frac{n}{2^n}x^n</math>.
# <math>\sum_{n=0}^\infty\frac{(2x+1)^n}{n^2}</math> near <math>x=-\frac12</math>.

'''Task 2.''' Solve the equation <math>(y')^2=1-y^2</math> with <math>y(0)=0</math> and <math>y'(0)>0</math> using power series up to and including the coefficient of <math>x^5</math>. Then compare your result with the Taylor expansion of the exact solution.

'''Task 3.''' Find the recurrence relation defining the power series solutions of the following equations:
# <math>y''-xy'-y=0</math> with <math>y(0)=1</math> and <math>y'(0)=0</math>.
# <math>y''-xy'-y=0</math> with <math>y(1)=1</math> and <math>y'(1)=0</math>.
# <math>(1-x)y''+y=0</math> with <math>y(0)=0</math> and <math>y'(0)=1</math>.

'''Task 4.''' In view of the theorem about convergence of power series solutions (Fuchs' theorem), give a lower bound on the radius of convergence of the series solution of the equation <math>(x^2-2x-3)y''+xy'+4y=0</math> near <math>x=0</math>.

'''Task 5.'''
# Find a recurrence relation satisfied by <math>a_n:=\begin{pmatrix}2n\\n\end{pmatrix}</math>.
# Find a differential equation satisfied by <math>y(x):=\sum_{n=0}^\infty\begin{pmatrix}2n\\n\end{pmatrix}x^n</math>.
# Solve that equation to determine <math>y(x)</math> in general, and <math>\sum_{n=0}^\infty\frac{1}{5^n}\begin{pmatrix}2n\\n\end{pmatrix}</math> in particular.

{{Template:12-267:Dror/Students Divider}}

[http://imgur.com/a/eeTWI#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]

12-267

2012-11-27T09:06:37Z

Vsbdthrsh:

__NOEDITSECTION__
__NOTOC__
{{12-267/Navigation}}
==Advanced Ordinary Differential Equations==
===Department of Mathematics, University of Toronto, Fall 2012===

{{12-267/Crucial Information}}

===Text===
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).

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* Vitali Kapovitch's 2007 classes: [http://www.math.toronto.edu/vtk/267/ Spring], [http://www.math.toronto.edu/vtk/267fall07/ Fall].

* Also previously taught by T. Bloom, C. Pugh, D. Remenik.

* My {{Pensieve Link|Classes/12-267/|12-267 notebook}}.

{{Template:12-267:Dror/Students Divider}}

[[User:Drorbn|Drorbn]] 06:36, 12 September 2012 (EDT): Material by [[User:Syjytg|Syjytg]] moved to [[12-267/Tuesday September 11 Notes]].

[http://imgur.com/a/OSx1U#0 Summary of techniques to solve differential equations ] [[User:Vsbdthrsh|Vsbdthrsh]]

[http://drorbn.net/index.php?title=12-267/Existence_And_Uniqueness_Theorem Fundamental Theorem and Proof from Lecture] [[User:Twine|Twine]]

[http://drorbn.net/index.php?title=12-267/Derivation_of_Euler-Lagrange Derivation of Euler-Lagrange from Lecture] [[User:Twine|Twine]]

[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]]

12-267 [http://math.hunter.cuny.edu/mbenders/cofv.pdf In-depth coverage of Calculus of Variations][[User:Simon1|Simon1]]

12-267 [http://highered.mcgraw-hill.com/sites/dl/free/007063419x/392340/Calculus_of_Variations.pdf A good summary of Calculus of Variations]
[[User:Simon1|Simon1]]

12-267 [http://imgur.com/a/1vy11#0 All class notes from September 10th to October 5th] [[User:Simon1|Simon1]]

12-267 [http://imgur.com/a/uLSlM Summary of Numerical Methods] [[User:Simon1|Simon1]]

[http://drorbn.net/index.php?title=Numerical_Methods Numerical Methods (wiki)] [[User:Twine|Twine]]

12-267 [http://imgur.com/a/tliYg Summary of Chapter 3 from the Textbook on Constant Coefficient Second Order ODEs] [[User:Simon1|Simon1]]

[http://drorbn.net/images/e/e6/Geometric_Interpretation_of_Lagrange_Multiplier.pdf Geometric Interpretation of Lagrange Multiplier] [[User:Mathstudent|Mathstudent]]

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]]

Handwritten notes by [[User:Ktnd3|Ktnd3]]:

* 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]]

* 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]]

[http://i.imgur.com/uTugV.jpg Quick guide: system of 1st order linear equations] [[User:Vsbdthrsh|Vsbdthrsh]]

Help of Inverse Matrix [http://mathworld.wolfram.com/MatrixInverse.html Matrix Inverse] [[User:Dongwoo.kang|Dongwoo.kang]]

12-267 [http://imgur.com/a/50sRR All class notes from October 5th to October 30th] [[User:Simon1|Simon1]]

[http://imgur.com/a/sZSYx#0 Quick guide: Power Series + ODE] [[User:Vsbdthrsh|Vsbdthrsh]]

12-267/Homework Assignment 6

2012-11-17T03:17:11Z

Vsbdthrsh:

{{12-267/Navigation}}

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.

'''Task 1.''' Draw the phase portraits for the following systems, near <math>(x,y)=(0,0)</math>:
# <math>\begin{cases} \dot{x}=2x+y \\ \dot{y}=-x+4y \end{cases}</math>.
# <math>\begin{cases} \dot{x}=4x-5y \\ \dot{y}=4x-4y \end{cases}</math>.
# <math>\begin{cases} \dot{x}=x-2y \\ \dot{y}=-2x+4y \end{cases}</math>.
# <math>\begin{cases} \dot{x}=-x+y \\ \dot{y}=-5x+3y \end{cases}</math>.
# <math>\begin{cases} \dot{x}=-5x+4y \\ \dot{y}=-8x+7y \end{cases}</math>.

'''Task 2.''' Draw the phase portrait of the system
<center>
<math>\begin{cases}\dot{x}=17+x-9y+\sin(2-2x-y+xy)\\\dot{y}=7+2x-5y+\cos(x-1)\end{cases}</math>
</center>
near the point <math>(x,y)=(1,2)</math>.

'''Task 3.''' Solve using diagonalization (one solution is enough):
# <math>v'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} e^t \\ t \end{pmatrix}</math>.
# <math>v'=\begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}v + \begin{pmatrix} -\cos t \\ \sin t \end{pmatrix}</math>.

'''Task 4.''' Assume <math>t>0</math>. For the following equation,
<center><math>tv'=\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v + \begin{pmatrix} 1-t^2 \\ 2t \end{pmatrix}</math></center>,
it is given that a solution of the homogeneous version is
<center><math>v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t + \begin{pmatrix}1\\3\end{pmatrix}t^{-1}</math>.</center>
Use "fundamental solutions" to find a solution of the full equation.

'''Task 5.''' (Not for grade). Find a quadratic differential equation whose phase portrait is as below.

[[Image:12-267-MonkeySaddleFlow.png|center|400px]]

'''Hint.''' "Monkey Saddle".

{{Template:12-267:Dror/Students Divider}}

[http://imgur.com/a/53nSl#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]

12-267/Homework Assignment 5

2012-11-07T21:53:59Z

Vsbdthrsh:

{{12-267/Navigation}}

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.

'''Task 1.''' Consider the following systems of equations:

<math>A:\begin{cases}
\dot{x}=x-2y & x(0)=3\\
\dot{y}=4y-2x & y(0)=1\\
\end{cases}</math>

<math>B:\begin{cases}
\dot{x}=x-5y & x(0)=3\\
\dot{y}=2x-5y & y(0)=1
\end{cases}</math>

<math>C:\begin{cases}
\dot{x}=y & x(0)=1 \\
\dot{y}=z & y(0)=2 \\
\dot{z}=-6x-11y-6z & z(0)=-1
\end{cases}</math>

# Write each one in a matrix form.
# Find the eigenvalues and eigenvectors of the resulting matrices.
# Diagonalize these matrices.
# Compute <math>e^{tA}</math> for each of those matrices.
# Solve these equations.

'''Task 2.'''
# 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>.
# Find an example for two matrices <math>A</math> and <math>B</math> for which <math>e^{A+B}\neq e^Ae^B</math>.

'''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>.

(Here, of course, <math>e^D:=\sum_{k=0}^\infty \frac{D^k}{k!}</math>).

{{Template:12-267:Dror/Students Divider}}

[http://imgur.com/a/txF9Z#0 Solutions] [[User:Vsbdthrsh|Vsbdthrsh]]

12-267/Homework Assignment 4

2012-11-01T07:46:24Z

Vsbdthrsh:

{{12-267/Navigation}}

This assignment is due at the tutorial on Tuesday October 16. 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.

'''Task 0.''' Identify yourself in the [[12-267/Class Photo|Class Photo]]!

'''Task 1.''' Find the general solution of the Euler-Lagrange equation corresponding to the functional <math>J(y)=\int_a^bf(x)\sqrt{1+y'^2}dx</math>, and investigate the special cases <math>f(x)=\sqrt{x}</math> and <math>f(x)=x</math>.

'''Task 2.''' Find the extrema the following functional <math>y\mapsto\int_0^1(y'^2+x^2)dx</math> subject to <math>\int_0^1y^2dx=2</math> and <math>y(0)=0</math> and <math>y(1)=0</math>. (An earlier version of this assignment had by mistake <math>y(1)=1</math>, which leads to much uglier numbers. If you already solved the problem with <math>y(1)=1</math>, you may submit either solution).

'''Task 3.''' Solve the "power line problem": Of all the curves <math>y</math> with <math>y(a)=A</math> and <math>y(b)=B</math> and with total arc-length <math>l</math>, find the one with the least potential energy <math>\int_a^by\sqrt{1+y'^2}dx</math>.

'''Task 4.''' Find a necessary condition for a function <math>y</math> satisfying <math>y(a)=A</math>, <math>y'(a)=A'</math>, <math>y(b)=B</math>, and <math>y'(b)=B'</math> to be an extremal of a functional of the form <math>y\mapsto\int_a^bF(x,y,y',y'')dx</math>.

'''Task 5.''' Find the curve <math>y</math> joining the points <math>(0,0)</math> and <math>(1,0)</math> and for which the integral <math>\int_0^1y''^2dx</math> is minimal, if <math>y'(0)=a</math> and <math>y'(1)=b</math>.

{{Template:12-267:Dror/Students Divider}}

'''Solution to Task 4.''' --[[User:Twine|Twine]] 17:54, 24 October 2012 (EDT)

Let <math>h(x)</math> be any function defined on <math>[a, b]</math> with <math>h'(a) = h(a) = h'(b) = h(b) = 0</math>. For y to be an extremal of the functional with the boundary constraints given, we must have that <math>\frac{d}{d\epsilon} J(y + \epsilon h) |_{\epsilon = 0} = 0</math> for any such <math>h(x)</math>.

<math>\frac{d}{d\epsilon} J(y + \epsilon h) |_{\epsilon = 0}</math>

<math>= \frac{d}{d\epsilon} \int_a^b F(x, y+\epsilon h, y'+\epsilon h', y'' +\epsilon h'')dx|_{\epsilon = 0}</math>

<math>= \int_a^b (F_1 \cdot 0 + F_2 \cdot h + F_3 \cdot h' + F_4 \cdot h'')dx|_{\epsilon = 0}</math>

<math>= \int_a^b (F_2 \cdot h - F_3' \cdot h - F_4' \cdot h')dx + F_3 \cdot h|_a^b + F_4\cdot h'|_a^b</math>

<math>= \int_a^b (F_2 \cdot h - F_3' \cdot h + F_4'' \cdot h)dx + F_4'\cdot h|_a^b</math>

<math>= \int_a^b (F_2 - F_3' + F_4'') \cdot h dx</math>

For this to be equivalent to 0 for any h defined above, we must have

<math>F_y - \frac{d}{dx}F_{y'} + \frac{d^2}{dx^2}F_{y''} = 0</math>

'''Solution to Task 5.''' --[[User:Twine|Twine]] 17:54, 24 October 2012 (EDT)

We use the result of 4. As in this case F is independent of x, y, and y', the equation reduces to

<math>\frac{d^2}{dx^2}2y'' = 0</math>

<math>y'''' = 0</math>

This has the solution <math>y = c_0 + c_1 x + c_2 x^2 + c_3 x^3</math>. We can use the constraint equations <math>y(0) = 0</math>, <math>y(1) = 0</math>, <math>y'(0) = a</math>, <math>y'(1) = b</math> to show that <math>y = (b+a)x^3 - (2a + b)x^2 + ax</math>. Hence, This is the only y for which the functional is extremal.

[http://imgur.com/a/j99mC#0 Solutions to task 1,2 and 3] [[User:Vsbdthrsh|Vsbdthrsh]]

12-267

2012-11-01T07:41:24Z

Vsbdthrsh:

12-267

2012-11-01T07:09:48Z

Vsbdthrsh:

__NOEDITSECTION__
__NOTOC__
{{12-267/Navigation}}
==Advanced Ordinary Differential Equations==
===Department of Mathematics, University of Toronto, Fall 2012===

{{12-267/Crucial Information}}

===Text===
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).

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* Vitali Kapovitch's 2007 classes: [http://www.math.toronto.edu/vtk/267/ Spring], [http://www.math.toronto.edu/vtk/267fall07/ Fall].

* Also previously taught by T. Bloom, C. Pugh, D. Remenik.

* My {{Pensieve Link|Classes/12-267/|12-267 notebook}}.

{{Template:12-267:Dror/Students Divider}}

[[User:Drorbn|Drorbn]] 06:36, 12 September 2012 (EDT): Material by [[User:Syjytg|Syjytg]] moved to [[12-267/Tuesday September 11 Notes]].

[http://imgur.com/a/OSx1U#0 Summary of techniques to solve differential equations ] [[User:Vsbdthrsh|Vsbdthrsh]]

[http://drorbn.net/index.php?title=12-267/Existence_And_Uniqueness_Theorem Fundamental Theorem and Proof from Lecture] [[User:Twine|Twine]]

[http://drorbn.net/index.php?title=12-267/Derivation_of_Euler-Lagrange Derivation of Euler-Lagrange from Lecture] [[User:Twine|Twine]]

[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]]

12-267 [http://math.hunter.cuny.edu/mbenders/cofv.pdf In-depth coverage of Calculus of Variations][[User:Simon1|Simon1]]

12-267 [http://highered.mcgraw-hill.com/sites/dl/free/007063419x/392340/Calculus_of_Variations.pdf A good summary of Calculus of Variations]
[[User:Simon1|Simon1]]

12-267 [http://imgur.com/a/1vy11#0 All class notes from September 10th to October 5th] [[User:Simon1|Simon1]]

12-267 [http://imgur.com/a/uLSlM Summary of Numerical Methods] [[User:Simon1|Simon1]]

[http://drorbn.net/index.php?title=Numerical_Methods Numerical Methods (wiki)] [[User:Twine|Twine]]

12-267 [http://imgur.com/a/tliYg Summary of Chapter 3 from the Textbook on Constant Coefficient Second Order ODEs] [[User:Simon1|Simon1]]

[http://drorbn.net/images/e/e6/Geometric_Interpretation_of_Lagrange_Multiplier.pdf Geometric Interpretation of Lagrange Multiplier] [[User:Mathstudent|Mathstudent]]

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]]

Handwritten notes by [[User:Ktnd3|Ktnd3]]:

* 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]]

* 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]]

[http://i.imgur.com/uTugV.jpg Quick quide: system of 1st order linear equations] [[User:Vsbdthrsh|Vsbdthrsh]]

12-267

2012-10-11T06:23:16Z

Vsbdthrsh:

12-267

2012-10-06T06:36:17Z

Vsbdthrsh:

12-267

2012-10-06T06:35:32Z

Vsbdthrsh: