Drorbn - User contributions [en]

12-267

2012-10-25T20:44:38Z

Mathstudent:

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==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]]

[http://drorbn.net/index.php?title=12-267/Euler%27s_Method Python example for Euler's Method] [[User:Twine|Twine]]

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

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

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

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

[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]]

12-267/Homework Assignment 2

2012-10-23T23:36:43Z

Mathstudent:

{{12-267/Navigation}}

This assignment is due at the tutorial on Tuesday October 2. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the your readers will be having 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.''' Solve the following differential equations:
# <math>x^2y^3+x(1+y^2)y'=0</math> (hint: try <math>\mu=x^\alpha y^\beta</math>).
# <math>dx+(\frac{x}{y}-\sin y)dy=0</math>.
# <math>(x^2+3xy+y^2)dx-x^2dy=0</math>.
# <math>\frac{dy}{dx}=\frac{2y-x+5}{2x-y-4}</math> (hint: consider trying <math>x_1=x+\alpha</math> and <math>y_1=y+\beta</math> for good <math>\alpha</math>, <math>\beta</math>).
# <math>y'=\frac{y^3}{1-2xy^2}</math> with <math>y(0)=1</math>.
# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.

'''Task 2.''' Let <math>M</math> and <math>N</math> be differentiable functions of <math>x</math> and <math>y</math>.
# Show that if <math>\frac{N_x-M_y}{xM-yN}</math> depends only on <math>xy</math>, then the differential equation <math>Mdx+Ndy=0</math> has an integrating factor of the form <math>\mu(xy)</math>, where <math>\mu</math> is a function of a single variable.
# Find a condition on <math>M</math> and <math>N</math> that would imply that the differential equation <math>Mdx+Ndy=0</math> would have an integrating factor of the form <math>\mu(x+y)</math>, where <math>\mu</math> is a function of a single variable.

'''Task 3.''' The equation <math>y'+p(x)y=q(x)y^n</math> is called a "Bernoulli Equation".
# Explain why you already know how to solve the Bernoulli equation when <math>n=0</math> and when <math>n=1</math>.
# Show that if <math>n\neq0,1</math>, then the substitution <math>v=y^{1-n}</math> reduces the Bernoulli equation to an equation you already know how to solve.
# Solve the equation <math>x^2y'+2xy-y^3=0</math> (for <math>x>0</math>).

'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.

[http://drorbn.net/index.php?title=Image:005.jpg Solution to HW2, page 1] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:006.jpg Solution to HW2, page 2] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:007.jpg Solution to HW2, page 3] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:008.jpg Solution to HW2, page 4] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 7] [[User:Mathstudent|Mathstudent]]

12-267/Homework Assignment 3

2012-10-23T23:29:26Z

Mathstudent:

{{12-267/Navigation}}

This assignment is due at the tutorial on Tuesday October 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 0.''' Identify yourself in the [[12-267/Class Photo|Class Photo]]!

'''Task 1.''' Let <math>\phi_n\colon X\to{\mathbb R}</math> be a sequence of functions defined on some set <math>X</math>, and suppose that some sequence <math>c_n</math> of non-negative reals is given such that for every <math>x\in X</math>, <math>|\phi_n(x)-\phi_{n+1}(x)|\leq c_n</math>. Suppose also that <math>\sum_{n=1}^\infty c_n</math> is finite. Prove that the sequence <math>\phi_n</math> is uniformly convergent.

'''Task 2.''' Find the extrema of the following functionals:
# <math>y\mapsto \int_0^1y'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.
# <math>y\mapsto \int_0^1yy'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.
# <math>y\mapsto \int_0^1xyy'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.
# <math>y\mapsto\int_a^b\frac{y'^2}{x^3}dx</math>.
# <math>y\mapsto\int_a^b(y^2+y'^2+2ye^x)dx</math>.
# <span style="color: red">Postponed!</span> <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)=1</math>.

'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

[http://drorbn.net/index.php?title=Image:012.jpg Solution to HW3, page 1] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:013.jpg Solution to HW3, page 2] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:014.jpg Solution to HW3, page 3] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:015.jpg Solution to HW3, page 4] [[User:Mathstudent|Mathstudent]]

12-267/Homework Assignment 3

2012-10-23T23:28:55Z

Mathstudent:

{{12-267/Navigation}}

This assignment is due at the tutorial on Tuesday October 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 0.''' Identify yourself in the [[12-267/Class Photo|Class Photo]]!

'''Task 1.''' Let <math>\phi_n\colon X\to{\mathbb R}</math> be a sequence of functions defined on some set <math>X</math>, and suppose that some sequence <math>c_n</math> of non-negative reals is given such that for every <math>x\in X</math>, <math>|\phi_n(x)-\phi_{n+1}(x)|\leq c_n</math>. Suppose also that <math>\sum_{n=1}^\infty c_n</math> is finite. Prove that the sequence <math>\phi_n</math> is uniformly convergent.

'''Task 2.''' Find the extrema of the following functionals:
# <math>y\mapsto \int_0^1y'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.
# <math>y\mapsto \int_0^1yy'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.
# <math>y\mapsto \int_0^1xyy'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.
# <math>y\mapsto\int_a^b\frac{y'^2}{x^3}dx</math>.
# <math>y\mapsto\int_a^b(y^2+y'^2+2ye^x)dx</math>.
# <span style="color: red">Postponed!</span> <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)=1</math>.

'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

[http://drorbn.net/index.php?title=Image:0012.jpg Solution to HW3, page 1] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:0013.jpg Solution to HW3, page 2] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:0014.jpg Solution to HW3, page 3] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:0015.jpg Solution to HW3, page 4] [[User:Mathstudent|Mathstudent]]

12-267/Homework Assignment 2

2012-10-23T23:26:37Z

Mathstudent:

12-267/Homework Assignment 1

2012-10-23T23:22:58Z

Mathstudent:

{{12-267/Navigation}}

This assignment is due at the tutorial on Tuesday September 25. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.

'''Question 1.''' Show that if <math>y=y_1(x)</math> is a solution of <math>y'+p(x)y=0</math>, and <math>y=y_2(x)</math> is a solution of <math>y'+p(x)y=g(x)</math>, then for any constant <math>c</math>, <math>y=cy_1+y_2</math> is a solution of <math>y'+p(x)y=g(x)</math>.

'''Question 2.''' Solve the following differential equations
# For <math>x>0</math>, <math>xy'+2y=\sin x</math>.
# <math>\frac{dy}{dx}=\frac{1}{e^y-x}</math> with <math>y(1)=0</math>; you may want to solve for <math>x</math> first.
# <math>xy'=\sqrt{1-y^2}</math>.
# <math>\frac{dy}{dx}=\frac{x-e^{-x}}{y+e^y}</math>.
# <math>xdx+ye^{-x}dy=0</math>, with <math>y(0)=1</math>.
# <math>\frac{dy}{dx}=\frac{ay+b}{cx+d}</math>, where <math>a,b,c,d</math> are arbitrary constants.
# <math>\frac{dy}{dx}=-\frac{ax+by}{bx+cy}</math>, where <math>a,b,c</math> are arbitrary constants.
# <math>0=(e^x\sin y + 3y)dx + (3(x+y)+e^x\cos y)dy</math>.

[http://drorbn.net/index.php?title=Image:001.jpg Solution to HW1, page 1] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:002.jpg Solution to HW1, page 2] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:003.jpg Solution to HW1, page 3] [[User:Mathstudent|Mathstudent]]
[http://drorbn.net/index.php?title=Image:004.jpg Solution to HW1, page 4] [[User:Mathstudent|Mathstudent]]

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2012-10-23T22:44:42Z

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2012-10-23T22:38:43Z

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2012-10-23T22:29:55Z

Mathstudent: Assignment 1 Page 1

Assignment 1 Page 1