S2 (use the end Solve "shifting theoren 11, ostat the problem y"-y' -2y = f(t), f(t)=...
use the Laplace tronsform to solve the IVP Y"-2y + y = f(t), y(O) = 1, 4! (0)=/ Where 0, +23 f(+) { +-3, +23 2 you may use the portial fraction decompositron! +373 +7 - 27/1 3 30 (3.1)? S2(5²-25+1) 32(3-1/2 5-12 arrive to the expression: but show steps to 32 (57-25+1)
6. Use the 1st shifting theorem 5. Y"-34 +2y = 2 to solve te at y + 4y + 4y y (0)=0, 4 (0) = 1 theorems to compute hifting
1. (5 points) Use a Laplace transform to solve the initial value problem: y' + 2y + y = 21 +3, y(0) = 1,5 (0) = 0. 2. (5 points) Use a Laplace transform to solve the initial value problem: y + y = f(t), y(0) = 1, here f(0) = 2 sin(t) if 0 Str and f(0) = 0 otherwise.
5. (11 points) Solve the following initial value problem, y" + 3y + 2y = g(t); y(0) = 0, 7(0) = 1/2 where g(t) = 38(t - 1) + uz(t) Type here to search
2. Use the Laplace Transform to solve the initial value problem y"-3y'+2y=h(t), y(O)=0, y'(0)=0, where h (t) = { 0,0<t<4 2, t>4
10. Use the Laplace transform to solve y" - 3y' +2y f(t), y(0)-0,'(0) 0, where (t)-(0 for 0 st < 4; for t 2 4 No credit will be given for any other method. (10 marks)
7. 49 + 2y(t) = 4t; y(0-3. Use Laplace transform to solve this problem.
5. (13 pts) Solve the following initial value problem: y" + 2y' + y-ul (t)o V"(0) 0. y(0) 0, (t-1), -(t1) cos
+ 44 7-8 Use the shifting theorems to compute. the inverse Laplace transforms L F (S) = 25-1 S2-45+20 7. (2 -55 e 8. F(S) = s²+65+10 and lifting theoren
Page 4 IV. Use the Laplace transform to solve the IVP y' - 2y + y = f(t), y(0) = 1, v/(0) = 1, where (10) 0, t <3 f(t) = t-3, 3 You may use the partial fraction decomposition 16–25+1) 5+(9–1 = (-) + ? + - , but you need to show all the steps needed to arrive to the expression - 022-28+1) in order to receive credit.