MAT 131 Final: MAT 131 SBU .Fall10 131Final Solutions

For each of the following, solve the initial value problem using the method of Laplace transforms. y"-3y'+2y = cos t y(0) = 0 y'(0) = -l y"'+4y"+y'-6y = -12 y(0) = 1 y'(0) = 4 y"(0) = -2 Use the convolution theorem to find the inverse Laplace transform of the following function. F(s) = 11s/(s2 +121)2 Solve the following equation. (y - x)dx + (x + y)dy = 0 Find the general solution to the following differential equation. d2y/dx2 + 4dy/dx + 5y = e-x -sin2x Obtain the general solution to the following differential equation dy/dx - y/x = xex

Consider the two-mass mechanical system shown on the next page. For simplicity, we adopt the following numerical values, m_1 = m_2 = 1, gamma_1 = gamma_2 = 1/2, y = 0, k_ 1 = k_2 = k =1. We also assume that: f_1(t) = f_2(t) = 0, y(0) = [y_1(0) y_2(0)] = [1 0], y'(0) = [y'_1(0) y'_2(0)]] [0 1]. With these assumptions, the initial value problem becomes: [y"_1(t) y"_2(t)] = [-1/2 0 0 -1/2] [y'_1(t) y'(2)] + [-2 1 1 -2] [y_1(t) y_2(t)], [y_1(0) y_2(0)] = [1 0], [y'_1(0) y'_2(0)] = [0 1] (a) Apply Laplace transforms to this problem, solving for Y(s) = [Y_1(s) Y_2(s)], the Laplace transform of y(t) = [y_1(t) y_2(t)]. Perform this computation "by hand". (b) Obtain the time-domain solution y (t), using Mathematica as you wish. (c) Use Mathematica to plot the two components of y(t) on the same graph on the time interval 0 lessthanorequalto t lessthanorequalto 10/

1 point) Consider the initial value problem dy y(0) = 7,-(0) =-5 dt dt dt Write down the Laplace transform of the left-hand side of the equation given the initial conditions (sA2-12s+36)Y-7s+5+84 Your answer should be a function of s and Y with Y denoting the Laplace transform of the solution y. Write down the Laplace transform of the right-hand side of the equation Your answer should be a function of s only. Next equate your last two answers and solve for Y. You have (-5/(s-6)+1/(s-3)+7s-89)/(s^2-12s+36) Your answer should be a function of s only. Finally take the inverse laplace transform of your third answer to obtain the solution of the given initial value problem. y(1) Your answer should be a function of t.