Homework 6
November 8, 2022
Due by 11:59 pm, Nov 14
Problem 1. Find the Inverse Laplace transform of the following function:
F (s) =
17
(s ? 3)(s2 + 2s + 2)
Problem 2. Find the Inverse Laplace transform of the following functions:
i)

2
s +9
F (s) = ln 2
s +1
ii)

1
F (s) = arctan
s
Hint: Look at F ? (s) and find L ?1 {F ? }. Suppose that f (t) = L ?1 (F ). Then L {f } = F and
L {tf (t)} = ?F ? . Use this to find f.
Problem 3. i) Find L {sin t ? t cos t} and L {t sin t}.
ii) Use Part i) to solve the following IVP:
ty ?? ? 2y ? + ty = 0,
y(0) = 1,
y ? (0) = 0
You won get a unique solution here, so expect a Ô somewhere in the answer.
Problem 4. Solve the following IVPs:
i)
y ?? + 6y ? + 9y = 1,
y(0) = ?1,
y ? (0) = 6
y(0) = 0,
y ? (0) = 0
ii)
y ?? + 3ty ? ? 6y = 1,
1

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