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Homework #7 Math 527

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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