12-267/Homework Assignment 3

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 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 Class Photo!

Task 1. Let $\phi _{n}\colon X\to {\mathbb {R} }$ be a sequence of functions defined on some set $X$, and suppose that some sequence $c_{n}$ of non-negative reals is given such that for every $x\in X$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi_n(x)-\phi_{n+1}(x)|\leq c_n} . Suppose also that $\sum _{n=1}^{\infty }c_{n}$ is finite. Prove that the sequence $\phi _{n}$ is uniformly convergent.

Task 2. Find the extrema of the following functionals:

1. $y\mapsto \int _{0}^{1}y'dx$ subject to $y(0)=0$ and $y(1)=1$.
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\mapsto \int_0^1yy'dx} subject to $y(0)=0$ and $y(1)=1$.
3. $y\mapsto \int _{0}^{1}xyy'dx$ subject to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0)=0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(1)=1} .
4. $y\mapsto \int _{a}^{b}{\frac {y'^{2}}{x^{3}}}dx$.
5. $y\mapsto \int _{a}^{b}(y^{2}+y'^{2}+2ye^{x})dx$.
6. Postponed! $y\mapsto \int _{0}^{1}(y'^{2}+x^{2})dx$ subject to $\int _{0}^{1}y^{2}dx=2$ and $y(0)=0$ and $y(1)=0$.

Task 3. A roach Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} 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 $R$'s life, help it find the fastest path from one point in the upper half plane Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{y>0\}} to another point in the upper half plane, assuming there is only one wall around, built along the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis $y=0$.

Dror's notes above / Student's notes below

Solution to Task 1. --Twine 18:11, 24 October 2012 (EDT)

Let $\sum _{n=1}^{\infty }c_{n}=S$. We have that $\lim _{n\rightarrow \infty }c_{n}=0$.

Let $\epsilon >0$.

$\exists N\in \mathbb {Z} ^{+}$ such that $\sum _{n=1}^{N}c_{n}>S-\epsilon$.

Choose such an N. Then $\sum _{n=N}^{\infty }c_{n}<\epsilon$.

Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall x \in X, \forall m > N, n>N} , we have $|\Phi _{n}(x)-\Phi _{m}(x)|<\epsilon$.

As this result is independent of our choice of x, we have by the cauchy criterion that $\Phi _{n}$ is uniformly convergent and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \rightarrow \infty} \Phi_n(x) = \Phi(x)} exists.

Solutions to HW3: Mathstudent

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