I am confused on how to go about all parts of six

MAT 22B: PROBLEM SET 4 3 Problem 5. (20 pts) Let uc(t) be the Heaviside step function at c E R. In this problem we study a third-order differential equation with a discontinuous external force. Solve the following initial value problem: y\"'(t)4y\"(t):4t+3u(t)ea-5a y(0):0. 2/(0120, y\"(0):4. Problem 6. (20 pts) Let [5(t) be the Dirac Delta distribution. In this problem we will study pendulums at rest which are suddenly affected by an impulse. (a) Let us consider a pendulum of angular frequency w = 2 with no friction at small oscillations, modeled by y\"(t) + 4y(t) = 0. Find the unique solution to the following Initial Value Problem: 9%) + 4W) = 5(t - 17), y(0) = 0, y'(0) = 0- Plot the solution y(t) you have obtained. Note that the initial conditions have the pendulum at absolute rest before the impulse at t : 1r is exerted. (b) How do the solutions of the Initial Value Problem y\"(t) + 4W) = \"r - 5(15 - s), 31(0) = 0, y'(0) = 0 vary for different values of \"y E R ? In particular, what are the dierences between positive and negative values of qr ? Qualitatively plot the solutions for qr = 10, 5,0,5, 10. (c) How do the solutions of the Initial Value Problem y\"(t) + 4W) : 5(1' - 6), 9(0) : 0, y'(0) = 0 differ from each other for different values of c E R+ ? Qualitatively plot the solutions for c = 1, 6, 12. (d) Find the unique solution to the following Initial Value Problem: y"(t) +4y(t) = 5(t 71') 5(t 2n), y(0) = 0, 33(0) = 0. Plot the solution y(t) you have obtained. Note that the initial conditions have the pendulum at absolute rest before the two impulses, at t = 7r and t = 271' are exerted. (e) Let us consider instead an inverted pendulum with no friction, modeled by we) ya) = 0. Let us study the effect of inserting an impulse after three seconds. Find the unique solution to the following Initial Value Problem: y"(t) - ya) = -20 - 5(t - 3), y(0) = 2, y'(0) = 4