Question

ve 2 for 2

Answer 1

Given differential equation y Prime + y = {2t - 2 for 1 lessthanorequalto t lessthanorequalto 2 -2t + 6 for 2 lessthanorequalto t lessthanorequalto 3 Auxiliary equation corresponding to the homogenous equation m^2 + 1 = 0 = > m = plusminus i Therefore complementary function y_C = C_1 cos t + C_2 sin t C_1, C_2 constants 0 for 1 lessthanorequalto t lessthanorequalto 2 particular integral, y_P = 1/D^2 + 1 (2t - 2) = (1 + D^2)^-1 (2t - 2) = (1 - D^2) (2t - 2) = 2t - 2 y = y_C + y_P = C_1 cos t + C_2 sin t + 2t - 2 y prime = -C_1 sin t + C_2 cos t + 2 Therefore y(0) = 0 = > C_1 - 2 = 0 = > C_1 = 2 y prime (0) = 0 = > C_2 + 2 = 0 = > C_2 = -2