Need to figure out how to work through these problems with the right solutions for practice exam.

1.(20pts) Use the Laplace transform to solve the given initial value problem. 2.(15pts) Seek power series solutions of the differential equation y" + 2y' - By = 0; y(0) = -2, (0) = 1 57" - 3xy' + 4y = 0 For full credit show all steps the way it was shown in the lecture videos and zoom about the point ro = 0. Find the recurrence relation. Show all steps the way it was meetings. shown in the lecture videos and zoom meetings. Do not find y and yz.3.(10pts) Find the inverse Laplace transform of the function 5.(13pts) Use the definition of the Laplace transform to find the Laplace transform F(s) for s > 2 of the function f(t) = e". For full credit show all steps the way it was shown in 4s - 7 F(s) = 32 + 8s + 21 the lecture videos and zoom meetings. 6.(12pts) Find the Laplace transform of the function /(t) = 4us(t) - 3(t+ 1) u.(t) + 26(t - 2). 4. (8pts) Find the inverse Laplace transform of the function Do not use the definition of the Laplace transform. se-25 F(s) = 82 + 3 Do not use the convolution theorem. MORE...7.(10pts) Determine a lower bound for the radius of convergence of series solutions about To = 4 Table of Laplace Transforms for the differential equation (x+ 2)(x' - 6x + 11)y" + 2ry' + 3y = 0. f (t) CLf (t)] = F(s) Show all steps as shown in the lecture videos and zoom meetings. (1) eat s - a ( 2 ) (3) sin at ( 4 ) S cos at 3 2 + 0 2 (5) got sin bt (s - a)2 + 62 (6) eat cos bt (s - a)'+ 62 (7) 8.(12pts) Find the inverse Laplace transform of the given function by using the convolution the- n! orem. Write all steps as shown in the lecture videos. Do not evaluate the convolution (s - a)n+1 (8) integral. F(s) = ($2 + 16)'