Laplace Transform Problems and Solutions for Mathematics 246, Test 2, Exams of Differential Equations

Download Laplace Transform Problems and Solutions for Mathematics 246, Test 2 and more Exams Differential Equations in PDF only on Docsity! November 6, 1989 Mathematics 246 Name. Test 2 USE ONE ANSWER SHEET (BOTH SIDES IF NECESSARY) FOR EACH QUESTION, FOR FULL CREDIT, SHOW YOUR WORK, 1. (a) Give the definition of "f is of exponential order as t > oo " (b) State and prove the theorem which gives the formula expressing the Laplace transform of the derivative of a function f in terms of the Laplace transform of the function and f(O). 2. <a> Calculate the Laplace transform of <S* (t 3ir/2) sin t <b> Calculate the Laplace transform of u3Trx=><t> sin t (c) Solve the following initial value problem y 1 ' - 4 y 1 + 5 y =. 2 £(t-2>, y<0) = 1, y'(0> = 1. 3 (a) Use the method of Variation of Parameters to solve y< • - 4 yi H- 4 y== ez« (b) Calculate the Wronskian of the two solutions of the homogeneous equation. 4. DO NOT USE THE LAPLACE TRANSFORM FOR EITHER OF THE FOLLOWING '. (a) Solve y<-»> + 3 y <=s> _3 y ti _ 7y i + & y = o (b) Solve y <3> + y 1 - 2 + sin x.