12-267/Homework Assignment 1: Difference between revisions

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'''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{ax+b}{cx+d}</math>, where <math>a,b,c,d</math> are constants.
# <math>\frac{dy}{dx}=-\frac{ax+by}{bx+cy}</math>, where <math>a,b,c</math> are constants.
# <math>0=(e^x\sin y + 3y)dx - (3x-e^x\sin y)dy</math>.

Revision as of 14:30, 17 September 2012

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. dbnvp Monday: Introduction and the Brachistochrone. dbnvp Tuesday: More on the Brachistochrone, administrative issues. Tuesday Notes. dbnvp Friday: Some basic techniques: first order linear equations.
2 Sep 17 dbnvp Monday: Separated equations, escape velocities. HW1. dbnvp Tuesday: Escape velocities, changing source and target coordinates, homogeneous equations. dbnvp Friday: Reverse engineering separated and exact equations.
3 Sep 24 dbnvp Monday: Solving exact equations, integration factors. HW2. dbnvp Tuesday: Statement of the Fundamental Theorem. Class Photo. dbnvp Friday: Proof of the Fundamental Theorem.
4 Oct 1 dbnvp Monday: Last notes on the fundamental theorem. HW3. dbnvp Tuesday Hour 1: The chain law, examples of variational problems. dbnvp Tuesday Hour 2: Deriving Euler-Lagrange. dbnvp Friday: Reductions of Euler-Lagrange.
5 Oct 8 Monday is thanksgiving. dbnvp Tuesday: Lagrange multiplyers and the isoperimetric inequality. HW4. dbnvp Friday: More Lagrange multipliers, numerical methods.
6 Oct 15 dbnvp Monday: Euler and improved Euler. dbnvp Tuesday: Evaluating the local error, Runge-Kutta, and a comparison of methods. dbnvp Friday: Numerical integration, high order constant coefficient homogeneous linear ODEs.
7 Oct 22 dbnvp Monday: Multiple roots, reduction of order, undetermined coefficients. dbnvp Tuesday: From systems to matrix exponentiation. HW5. Term Test on Friday.
8 Oct 29 dbnvp Monday: The basic properties of matrix exponentiation. dbnvp Tuesday: Matrix exponentiation: examples. dbnvp Friday: Phase Portraits. HW6. Nov 4 was the last day to drop this class
9 Nov 5 dbnvp Monday: Non-homogeneous systems. dbnvp Tuesday: The Catalan numbers, power series, and ODEs. dbnvp Friday: Global existence for linear ODEs, the Wronskian.
10 Nov 12 Monday-Tuesday is UofT November break. HW7. dbnvp Friday: Series solutions for [math]\displaystyle{ y'=f(x,y) }[/math].
11 Nov 19 dbnvp Monday: [math]\displaystyle{ \pi }[/math] is irrational, more on the radius of convergence. dbnvp Tuesday (class): Airy's equation, Fuchs' theorem. dbnvp Tuesday (tutorial): Regular singular points. HW8. dbnvp Friday: Discussion of regular singular points..
12 Nov 26 dbnvp Monday: Frobenius series by computer. Qualitative Analysis Handout (PDF). dbnvp Tuesday: The basic oscillation theorem. Handout on the Frobenius Method. HW9. dbnvp Friday: Non-oscillation, Sturm comparison.
13 Dec 3 dbnvp Monday: More Sturm comparisons, changing the independent variable. dbnvp Tuesday: Amplitudes of oscillations. Last class was on Tuesday!
F1 Dec 10
F2 Dec 17 The Final Exam (time, place, style, office hours times)
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The information below is preliminary and cannot be trusted! (v)

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

Question 2. Solve the following differential equations

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