MA 238: Test 3 Solutions - Laplace Transform Problems - Prof. Shangbing Ai, Exams of Differential Equations

Download MA 238: Test 3 Solutions - Laplace Transform Problems - Prof. Shangbing Ai and more Exams Differential Equations in PDF only on Docsity! MA 238: TEST 3 (12/6/04) INSTRUCTIONS: Write your name on each answer sheet and show all your work on the problems. 1. Use the Laplace transform to solve the initial value problem x′′ + 8x′ + 15x = 0, x(0) = 2, x′(0) = −3. 2. Use the Laplace transform to solve the initial value problem y′′ + y = t, y(0) = 0, y′(0) = 0. 3. (a) Find L−1{ 3 s2 + 6s + 18 }. (b) Use the Laplace transform and the convolution theorem to solve the initial value problem x′′ + 6x′ + 18x = 3 cos 2t, x(0) = 1, x′(0) = 0. 4. Use the Laplace transform and the convolution theorem to find a formula for the solution of the initial value problem y′′ + 9y = f(t), y(0) = 2, y′(0) = 1. 5. (a) Let f(t) = −1 if t < 1 and f(t) = t if t ≥ 1. Find L{f(t)}. (b) Use the second translation theorem to find L−1{F (t)} where F (s) = e −2s s− 1.