(1) Given IVP y" + 25y =d(t7) + 256(t ), y(0) = 1,y' (0) = 0, Tk (a) Solve the given IVP. Simplify your answers using trigonometric formulas for sin / cos(t). (b) Graph the obtained solution. 8 (2) Use the convolution theorem to find the inverse Laplace transform of function (s +9) (s +25) (3) (a) Express the solution of the given initial value problem in terms of a convolution integral: y" - 10y' +29y = g(t), y(0) = 0, y'(0) = -2. (1) (b) Find the solution of the same initial value problem (1) using the method of variation of parameter. Show that your answer coincides with the answer obtained in item (a). (4) Find the power series solution of the initial value problem y" (x) + xy'(x) + 2xy = 0, y(0)= 1, y' (0) = 0. (note that since the initial condition are given for xo = 0, then you should look for the power series solution around ro = 0). (5) Determine a low bound the radius of convergence of the equation ries (x- 6x +34)y" + (x+2)y' + sin ry = 0 about each of the following points to: (a) xo = -3 (b) xo = -1; (c) xo = 3.