MAP 2302 Differential Equations Summer 2017 Sample Exam 3 No calculating device will be allowed. For the exam, you can only use a pen/pencil. A Table for Laplace Transforms will be provided 1 2 . 1. Find the general solution to y 00 + 6y 0 + 9y = e3t . 1+t 1'. Find the general solution to y 00 + 4y 0 + 4y = e2t ln t. 2. Find the general solution to (t 2)2 y 00 (t) + 7(t 2)y 0 (t) + 5y(t) = 0, 2'. Find the general solution to (t 2)2 y 00 (t) 3(t 2)y 0 (t) + 6y(t) = 0. 3. Suppose L {f (t)}(s) = 1 , f (0) = 1, and f 0 (0) = 1. Determine the Laplace s+1 transform of e3t f (t) + tf (t) f 00 (t) + f 0 (2t) + e2t sin2 (4t) 4. Determine L 1 {F (s)}(t) if F (s) = s2 + 3s + 4 . (s + 2)3 4'. Determine L 1 {F (s)}(t) if F (s) = s s2 + 3s + 4 . (s + 2)3 4\". Determine L 1 {F (s)}(t) if F (s) = 1 2 3s3 + s 3 . s4 + 2s3 + 3s2 Ansatz y = v1 y1 + v2 y2 for ay 00 + by 0 + cy = f satisfies y1 v10 + y2 v20 = 0 and y10 v10 + y20 v20 = f /a. Characteristic equation for the Cauchy-Euler equation at2 y 00 + bty + cy = f is ar2 + (b a)r + c = 0. 1