University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2022
Views:

229

Pages:

1
Author:

uteromalaxating

Review Questions: Laplace Transforms (h) y 00 + y = tions both 0. 6. Miscellaneous Problems: Z ∞ π sin(3t)δ(t − ) dt (a) Evaluate: 2 0 (b) Evaluate, using Laplace transforms: sin(t)∗t (a) f (t) = 3, 0≤t≤2 6 − t, 2 < t (c) Use the table to find an expression for L(ty 0 ). Use this to solve: (b) f (t) = e−t , 0 ≤ t ≤ 5 −1, t > 5 y 00 + 3ty 0 − 6y = 1, y(0) = 0, y 0 (0) = 0 7. Characterize ALL solutions to y 00 +4y = u1 (t−1), y(0) = 1, y 0 (0) = 1. 2. Check your previous answer by rewriting f (t) using the step (or Heaviside) function, and use the table to compute the corresponding Laplace transform. 8. Define δ(t − c) as a limit of “regular” functions. 9. If y 0 (t) = δ(t − c), what is y(t)? 3. Determine the Laplace transform: Rc 10. Show that L(g(t+c)) = ecs (G(s)− 0 e−st g(t) dt). This might be useful, as mentioned in the notes on inverting f (t)uc (t). (a) t2 e−9t (b) e2t − t3 − sin(5t) (c) u5 (t)(t − 5)4 (d) e3t sin(4t) (e) et δ(t − 3) (f) t2 u4 (t) 4. Find the inverse Laplace transform: 2s − 1 − 4s + 6 s2 + 16s + 9 (b) (s + 1)(s + 3)(s − 2) 7 (c) (s + 3)3 (a) δ(t − 2kπ), with initial condi- k=1 1. Use the definition of the Laplace transform to determine L(f ): ∞ X s2 e−2s (4s + 2) (s − 1)(s + 2) 3s − 2 (e) 2s2 − 16s + 10 e−2s (f) 2 s + 2s − 2 (d) 5. Solve the given initial value problems using Laplace transforms: (a) y 00 − 7y 0 + 10y = 0, y(0) = 0, y 0 (0) = −3 (b) y 00 + 6y 0 + 9y = 0, y(0) = −3, y 0 (0) = 10 (c) y 00 +2y 0 +2y = t2 +4t, y(0) = 0, y 0 (0) = −1 (d) y 00 + 9y = 10e2t , y(0) = −1, y 0 (0) = 5 (e) y 00 − 2y 0 − 3y = u1 (t), y(0) = 0, y 0 (0) = −1 (f) y 00 − 4y 0 + 4y = t2 et , y(0) = 0, y 0 (0) = 0 (g) y 00 + 4y = δ(t − π2 ), y(0) = 0, y 0 (0) = 1 1

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