I need help setting up and solving 3A, 3B, 3C and 3D

2 DUE TO FRIDAY NOV 8 AT 10:00AM Problem 2. Show that the Laplace transform satisfies the following properties: (a) L(ect . f(t) ) (s) = [(f(t)) (s - c), where c E R. ( b ) L (uc (t ) . f (t - c ) ) ( s ) = e-es . L (f ( t ) ) ( s ) , where ce R . ( c ) L ( tn . f (t) ) (s ) = ( - 1) ". [(f (t) ) (n) (s ). Problem 3. (20 pts) Solve the following four Initial Value Problems using the method of the Laplace transform: (a) y" (t) + gy(t) = ett, y(0) = 1, y'(0) =0, (b) y" (t) + 6y'(t) + 6y(t) = sin(2t), y(0) = 0, y'(0) = 1. (c) y(4) (t ) - 4y(t) = 0, y(0) = 1, y' (0) =1, y"(0) = -2, y''(0) = 0. (d) y(10) (t) - 15y (8) (t) + 85y(6) (t) - 225y(4)(t) + 274y(2)(t) - 120y(t) = 0, with the following set of initial conditions y(0) = 0, y'(0) = 0, y(2)(0) = 0, y(3) (0) = 0, y(4) (0) = 0, y (5) (0) = 0, y (0) = 0, y? (0) = 0, y(8) (0) = 0, y(9) (0) = 1. Hint: The polynomial $10 - 15s8(t) + 856 - 225s4 + 274s2 - 120 factorizes as ($2 - 1) ($2 - 2) ($2 - 3) ($2 - 4)($2 - 5). Problem 4. (20 pts) Consider the following Initial Value Problem: y" (t ) + ty' (t) - 2y(t) = 2, y(0) = 0, y'(0) = 0. This is a non-homogeneous linear second-order differential equation with non-constant coefficients and not of Euler type. Note that we have not seen a previous method to solve this, and thus Laplace transform is essentially the only method at this point. (a) Write the Laplace transform of the Initial Value Problem above. (b) Find a closed formula for the Laplace transform L(y(t)). Hint: You will have to solve a first-order differential equation on L(y(t)). (c) Find the unique solution y(t) to the Initial Value