## Description

Homework #7 Math 527

Follow the usual instructions on homework submission: Be clear, legible, and organized.

Write on loose-leaf paper. Staple together in the left-hand corner, write your name,

section #, Math 527 HW6, and date in the upper-right-hand corner.

Problems 1-4: Find the Laplace transform or inverse Laplace transform as indicated.

1. L {(3t + 1)U (t − 1)}

2. L

e

2t

(t − 1)2

3. L −1

2s + 5

s

2 + 6s + 34

4. L −1

se−πs/2

s

2 + 4

Problem 5-6: Express the function f(t) in terms of the Heaviside function U (t − a)

and then find the Laplace transform L {f(t)}.

5. f(t) = (

sin t 0 ≤ t < 2π

0 2π ≤ t

6. f(t) = (

0 0 ≤ t < 1

t

2 1 ≤ t

Problem 7-10: Use Laplace transforms to solve the initial-value problems. If you like,

generate plots of the solutions using numerical software like Matlab, Python, or Julia.

7. y00 − 5y

0 + 6y = U (t − 1), y(0) = 0, y0

(0) = 1

8. y0 + 2y = f(t), y(0) = 0, where f(t) = (

t 0 ≤ t < 1

0 1 ≤ t

9. y00 + 2y

0 + y = f(t), y(0) = 0, y0

(0) = 1, where f(t) = (

0 0 ≤ t < 3

2 3 ≤ t

10. y00 + 4y

0 + 5y = δ(t − 2π), y(0) = y

0

(0) = 0