1.

Solve the initial value problem d df + 2y = 15sin(t) + 10 cos(t) with (0) = 8. y= . Reminder: To find the anti-derivative of " sin(u), the trick is to do integration by parts twice. Solve the initial value problem dt - 5x = cos(4t) with a (0) = 5. (t) =A. Let g(t) be the solution of the initial value problem dy +3y=0 d + y with (0) = 1. Find g(t). g(t) =| | B. Let f(t) be the solution of the initial value problem dy B r + 3y = exp(5t) with 4(0) = 1/8. Find f(t). &= | C. Find a constant so that k(t) = f(t) + cg(t) solves the differential equation in part B and k(0) = 16. e | d Solve the following differential equation. d_y =2 3yandy=1whenz =0 X = Hint: Recognize this as a first-order linear differential equation and follow the general method for solving these and use the initial conditions to find the integration constant