I’m working on a differential equations writing question and need an explanation to help me learn.

Write out solutions to all of the following problems on paper or type them using a mathematical typesetting environment. All written work must be completed in a clear, organized, and thorough way.

Differential Equations Write out solutions to all of the following problems on paper or type them using a mathematical typesetting environment. All written work must be completed in a clear, organized, and thorough way. Written work that cannot be readily understood will receive reduced or no credit 1. Use the integral definition of the Laplace Transform to find L{te4t }. 2. Use results derived in the Section 4-2 lecture to compute the Laplace Transform of each of the following functions. (That is, you do NOT need to use the integral definition of the Laplace Transform on this problem.) a) f(t) = 2 sin(t) − 3 sin(4t) + t 3 3 − e 5t b) f(t) = e −t (cos(2t) − 1.5 sin(2t)) 3. Compute the Laplace Transform of the following differential equation. Once you have computed the transform of each individual term, you can stop – you do not need to do any additional simplification. y 00 − y 0 − 2y = te2t − t 2 e −t 4. Find f(t) for each Laplace Transform F(s) given below. a) F(s) = 2 s − 6 b) F(s) = 2/3 s 2 + 6 c) F(s) = 4 (s − 1)2 d) F(s) = 3s s 2 − s − 6 e) F(s) = 2s + 2 s 2 + 2s + 5 5. Use Laplace Transforms to solve the following initial value problem. y 00 + 6y 0 + 34y = 0 with y(0) = 3 and y 0 (0) = 1 6. Write the following function in terms of Heaviside step functions. f(t) =    cos(2t) 0 ≤ t < 2π 0 2π ≤ t 7. Find the Laplace Transform of each of the following functions. a) g(t) = t − u3(t)(t − 3) + u3(t) sin(π(t − 3)) b) g(t) = u1(t)t 2 − uπ(t) cos(t) 8. For each F(s) given below, find f(t). a) F(s) = e −2s s 2 + s − 2 b) F(s) = 2 − e −s s 2 − 4s + 5 c) F(s) = e −5s − e −10s 5s 2 (s 2 + 4) 9. Use Laplace Transforms to solve the following differential equation: y 00 + 4y = f(t) with y(0) = 0, y 0 (0) = 0, and f(t) is the function defined in problem 6 above.

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