## Description

Homework 5: Math 22B

(20 points)

Instructions : Solve all problems. Print out your solutions when computer results are asked for,

work neatly, label your plots, show your work. Staple your homework together with your name

on it. A random subset of the problems will be graded. You are encouraged to work in groups,

but everyone must do their own write up.

Warning : Unstapled homework with multiple pages is minimum -5 out of 20 points and if a page

is lost from an unstapled homework the default assumption will be it was not turned in.

Reading Assignment : Read Boyce and Diprima Chapter 3.5-3.9

Problem 1 : [Programming in Matlab – section 2.7] Write a simple program in Matlab to solve the

initial value problem for t ∈ [0, 5].

dy

dt = −ty + 0.5y

3

, y(0) = 1

using the Euler Method and an appropriately chosen time step. Make sure to include the code

with your homework.

Problem 2 : Solve the given IVP and use Matlab to plot your solution (just plotting, you don’t

have to solve it numerically).

y

00 + 3y = 0, y(0) = −2, y0

(0) = 3

Problem 3 : Solve the IVP

2y

00 + 3y

0 − 2y = 0, y(0) = 1, y0

(0) = −β

and (b) plot the solution when β = 1 and find the coordinates (t0, y0) of the minimum point of

the solution in this case.

Problem 4 : Verify that y1 = e

t and y2 = tet are solutions of y

00 − 2y

0 + y = 0. Do they form a

fundamental set of solutions?

Problem 5 :

Problem 6 : Use the method of order reduction to find a second solution given y1 = t

2

, t 0 and

t

2

y

00 − 4ty0 + 6y = 0

Problem 7 : Use the method of order reduction to find a second solution given y1 = e

x

, x 1 and

(x − 1)y

00 − xy0 + y = 0

Problem 8 :Find the general solution to y

00 − 2y

0 − 10y = 0

Problem 9 : Find the particular solution

y

00 + 2y

0 + 2y = 0, y(π/4) = 0, y0

(π/4) = −2

Problem 10 : Find the particular solution

y

00 − 6y

0 + 9y = 0, y(0) = 0, y0

(0) = 2