12-267/Homework Assignment 1

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!

In Preparation

The information below is preliminary and cannot be trusted! (v)

Question 1. Show that if $y=y_{1}(x)$ is a solution of $y'+p(x)y=0$, and $y=y_{2}(x)$ is a solution of $y'+p(x)y=g(x)$, then for any constant $c$, $y=cy_{1}+y_{2}$ is a solution of $y'+p(x)y=g(x)$.

Question 2. Solve the following differential equations

1. For $x>0$, $xy'+2y=\sin x$.
2. ${\frac {dy}{dx}}={\frac {1}{e^{y}-x}}$ with $y(1)=0$; you may want to solve for $x$ first.
3. $xy'={\sqrt {1-y^{2}}}$.
4. ${\frac {dy}{dx}}={\frac {x-e^{-x}}{y+e^{y}}}$.
5. $xdx+ye^{-x}dy=0$, with $y(0)=1$.
6. ${\frac {dy}{dx}}={\frac {ay+b}{cx+d}}$, where $a,b,c,d$ are arbitrary constants.
7. ${\frac {dy}{dx}}=-{\frac {ax+by}{bx+cy}}$, where $a,b,c$ are arbitrary constants.
8. $0=(e^{x}\sin y+3y)dx+(3(x+y)+e^{x}\cos y)dy$.