## Description

Homework 6: 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 6.1 – 6.2

Problem 1 : Find the general solution using the method of undetermined coecients (3.5.5)

y00 + 9y = t

2

e3t + 6

Problem 2 : Find the general solution using the method of undetermined coecients (3.5.8)

y00 + 2y0 + y = 2et

Problem 3 : Find the general solution using the method of undetermined coecients (3.5.18)

y00 + 2y0 + 5y = 4et cos(2t), y(0 = 1, y0

(0) = 0

Problem 4 : (3.6.1) Use the variation of parameters method from section 3.6 to solve for the general

solution

y00 5y0 + 6y = 2et

Check your answer by the method of undetermined coecients.

Problem 5 : (3.6.15) Verify that y1 =1+ t and y2 = et satisfy the di↵erential equation

ty00 (1 + t)y0 + y = t

2

e2t

, t 0

then find the general solution using the variation of parameters method to find the particular

solution.

Problem 6 : (3.7.1) Find R, !0, to write u = 3 cos(2t) + 4 sin(2t) = R cos(!0t ).

Problem 7 : (3.7.2) Find R, !0, to write u = 2 cos(⇡t) 3 sin(⇡t) = R cos(!0t ).

Problem 8 : (3.7.26) The position of a certain spring satisfies

mu00 + u0 + ku = 0, u(0) = u0, u0

(0) = v0

Solve the initial value problem, assuming 2 < 4km. Then determine R in terms of m, , k, u0, v0

and write the solution in the form

u(t) = Ret/2m cos(µt )

Problem 9 : (3.7.29) The position of a certain spring satisfies

u00 +

1

4

u0 + 2u = 0, u(0) = 0, u0

(0) = 2

(a) Find the solution and (b) using MATLAB plot u vs. t and u0 vs. t on the same set of axis.

Problem 10 : (3.8.19) Consider the vibrating system

u00 + u = 3 cos(!t), u(0) = 1, u0

(0) = 1

(a) Find the solution for ! 6= 1, (b) using MATLAB plot u vs. t for ! = 0.7.