Question

12 which average distance does I travel Tn the laboratory Traie Question 5 Solve the following 2nd order linear ODE 2d2y(x) + dy (x)--y(x) = 2x2 dx Recall that this requires to first determine the (two) solutions of the associate homoge- neous equation: I121 and to then find a particular solution of the full equation The complete solution is then the superposition of the two terms solving the homogenous equation and the particular solution. a 53219010 581621.jp58637727 225536.Jp 53637727 225536...j,pg

Answer 1

The auxiliary equation is 2 m^2 + m - 1 = 0. m = -1 plusminus squareroot 1 + 8/4 = -1 plusminus 3/4 = -4/4 or 2/h = -1 or 1/2 Therefore The complementary function is y_c = c_1 e^- x + c_2 e%1/2 x Particular integral = 1/D^2 + D - 1 2 x^2 = 1/ - 1 (1 - (2D^2 + D)) 2 x^2 = -(1 - (2D^2 + D))^- 1 2x^2 = -(1 + 2D^2 + D + (2D^2 + D)^2 + .....) 2x^2 = -(2 x^2 + (2D^2 + D) 2x^2 + D^2 2x^2) = -(2x^2 + 8 + 4x + 4) = -2x^2 - 4x + 12. Therefore general solution is y_c + P.I = c_1 e^- x + c_2 e^1/2 x + 2 x^2 - 4x - 12