12-267/Homework Assignment 4

DBN: Classes: 2012-13: 12-267 / Navigation Additions to this web site no longer count towards good deed points. # Week of... Notes and Links 1 Sep 10 About This Class.  Monday: Introduction and the Brachistochrone.  Tuesday: More on the Brachistochrone, administrative issues. Tuesday Notes.  Friday: Some basic techniques: first order linear equations. 2 Sep 17  Monday: Separated equations, escape velocities. HW1.  Tuesday: Escape velocities, changing source and target coordinates, homogeneous equations.  Friday: Reverse engineering separated and exact equations. 3 Sep 24  Monday: Solving exact equations, integration factors. HW2.  Tuesday: Statement of the Fundamental Theorem. Class Photo.  Friday: Proof of the Fundamental Theorem. 4 Oct 1  Monday: Last notes on the fundamental theorem. HW3.  Tuesday Hour 1: The chain law, examples of variational problems.  Tuesday Hour 2: Deriving Euler-Lagrange.  Friday: Reductions of Euler-Lagrange. 5 Oct 8 Monday is thanksgiving.  Tuesday: Lagrange multiplyers and the isoperimetric inequality. HW4.  Friday: More Lagrange multipliers, numerical methods. 6 Oct 15  Monday: Euler and improved Euler.  Tuesday: Evaluating the local error, Runge-Kutta, and a comparison of methods.  Friday: Numerical integration, high order constant coefficient homogeneous linear ODEs. 7 Oct 22  Monday: Multiple roots, reduction of order, undetermined coefficients.  Tuesday: From systems to matrix exponentiation. HW5. Term Test on Friday. 8 Oct 29  Monday: The basic properties of matrix exponentiation.  Tuesday: Matrix exponentiation: examples.  Friday: Phase Portraits. HW6. Nov 4 was the last day to drop this class 9 Nov 5  Monday: Non-homogeneous systems.  Tuesday: The Catalan numbers, power series, and ODEs.  Friday: Global existence for linear ODEs, the Wronskian. 10 Nov 12 Monday-Tuesday is UofT November break. HW7.  Friday: Series solutions for $y'=f(x,y)$. 11 Nov 19  Monday: $\pi$ is irrational, more on the radius of convergence.  Tuesday (class): Airy's equation, Fuchs' theorem.  Tuesday (tutorial): Regular singular points. HW8.  Friday: Discussion of regular singular points.. 12 Nov 26  Monday: Frobenius series by computer. Qualitative Analysis Handout (PDF).  Tuesday: The basic oscillation theorem. Handout on the Frobenius Method. HW9.  Friday: Non-oscillation, Sturm comparison. 13 Dec 3  Monday: More Sturm comparisons, changing the independent variable.  Tuesday: Amplitudes of oscillations. Last class was on Tuesday! F1 Dec 10 F2 Dec 17 The Final Exam (time, place, style, office hours times) Register of Good Deeds Add your name / see who's in!

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 Class Photo!

Task 1. Find the general solution of the Euler-Lagrange equation corresponding to the functional $J(y)=\int _{a}^{b}f(x){\sqrt {1+y'^{2}}}dx$, and investigate the special cases $f(x)={\sqrt {x}}$ and $f(x)=x$.

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

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

Task 4. Find a necessary condition for a function $y$ satisfying $y(a)=A$, $y'(a)=A'$, $y(b)=B$, and $y'(b)=B'$ to be an extremal of a functional of the form $y\mapsto \int _{a}^{b}F(x,y,y',y'')dx$.

Task 5. Find the curve $y$ joining the points $(0,0)$ and $(1,0)$ and for which the integral $\int _{0}^{1}y''^{2}dx$ is minimal, if $y'(0)=a$ and $y'(1)=b$.

Dror's notes above / Student's notes below

Solution to Task 4. --Twine 17:54, 24 October 2012 (EDT)

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

${\frac {d}{d\epsilon }}J(y+\epsilon h)|_{\epsilon =0}$

$={\frac {d}{d\epsilon }}\int _{a}^{b}F(x,y+\epsilon h,y'+\epsilon h',y''+\epsilon h'')dx|_{\epsilon =0}$

$=\int _{a}^{b}(F_{1}\cdot 0+F_{2}\cdot h+F_{3}\cdot h'+F_{4}\cdot h'')dx|_{\epsilon =0}$

$=\int _{a}^{b}(F_{2}\cdot h-F_{3}'\cdot h-F_{4}'\cdot h')dx+F_{3}\cdot h|_{a}^{b}+F_{4}\cdot h'|_{a}^{b}$

$=\int _{a}^{b}(F_{2}\cdot h-F_{3}'\cdot h+F_{4}''\cdot h)dx+F_{4}'\cdot h|_{a}^{b}$

$=\int _{a}^{b}(F_{2}-F_{3}'+F_{4}'')\cdot hdx$

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

$F_{y}-{\frac {d}{dx}}F_{y'}+{\frac {d^{2}}{dx^{2}}}F_{y''}=0$

Solution to Task 5. --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

${\frac {d^{2}}{dx^{2}}}2y''=0$

$y''''=0$

This has the solution $y=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}$. We can use the constraint equations $y(0)=0$, $y(1)=0$, $y'(0)=a$, $y'(1)=b$ to show that $y=(b+a)x^{3}-(2a+b)x^{2}+ax$. Hence, This is the only y for which the functional is extremal.