Math 269A: Assignment 1

Assigned Monday Oct 8, due Monday Oct 15

Theory

[1](a) Give the general solution to y'' + ay' + by = 0.

(b) Describe the qualitative behavior of the solution when a > 0, a = 0, and a < 0.

(c) Describe the qualitative behavior of the solution when a2- 4b ³ 0 and a2- 4b < 0.


[2](a)Write the equation in problem [1] as a first order system and then give the general solution to this system.

(b) Let l e C be an eigenvalue of this system; characterize the qualitative behavior of the solutions of this system for the three cases Re(l) > 0, Re(l) = 0, and Re(l) < 0.

(c) Let l e C be an eigenvalue of this system; characterize the qualitative behavior of the solutions of this system for the cases Im(l) = 0, and Im(l) ¹ 0.


[3] Solve the following nonlinear ODE:

ODE

 

Computation

[5](a) Implement Euler's method and use it to solve

ODE

(b) Give the solution at t = 2.0 using timesteps dt = 0.1, 0.05, 0.025.
(c) Use the exact solution from [3] and numerical solution from [5]a to find the error in the solution at t = 2.0 using timesteps dt = 0.1, 0.05, 0.025.

Note: Use Matlab to do the computational problem. You can start by downloading and running the M-file EulerSample.m. This file uses Euler's method to solve dy/dt = -2y with initial condition y(0) = 1. The function F(y,t) = -2y is specified in the file F.m. After you figure out how to use matlab M-files, you can then modify these files so that they solve the problem in [5](a).

EulerSample.m
F.m

What You Should Turn In