Laplace Transforms:
1. Section 50: Problem 5.
2. Section 50: Problem 6.
3. Section 51: Problem 1.
4. Section 52: Problem 2a.
5. Section 52: Problem 5.
6. Prove the following:
(a) (f ∗ g)(t) = (g ∗ f)(t).
(b) If f and g are piecewise continuous and of exponential order on [0, ∞), then (f ∗ g)(t) is of exponential order on [0, ∞).
7. Prove the second translation theorem (in time): If F (s) = L{f(t)}(s), then
L{ua(t)f(t − a)}(s) = e−asF (s) (a ≥ 0).
Here ua(t) is the unit step function defined as ua(t) = 1, if t ≥ a, and = 0 if t < a.
8. Solve the following IVP using the Laplace transform method:
y′′ − y = t − 2
with y(2) = 3 and y′(2) = 0.
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