Question

(4 PLANTz une solución en oF 1CIENTES INDET22 para las siguientes ecuaciones PLANTER NO haga ninguna evaǐยocion numerica) (A) y" + 4y, + 5 y 3cos x (B) y" + 4y, + 5 y =2x3 -2 en x 3 sen 3 x (D) y" + 4y' + 5 y

Show a solution in undetermined coefficients for the following equations (only show, DO NOT make any numerical calculations)

Answer 1

A.E is m^2 + 4m + 5 = 0 m = -4 plusminus squareroot 4^2 - 4 (1) 5/2 (1) = -4 plusminus squareroot -4/2 = -4 plusminus 2i/2 = -2 plusminus i therefore complementary function y_c = e^-2x (A cos x + B sin x) RHS: R(x) = 3 cos x fundamental set of cos x is {sin x, cos } As sin x & cos x one not a member in y_c (i.e they are not solution of the corresponding homogeneous differential equation because particular is of the form y_p = c_1 sin x + c_2 cos x where c_1 & c_2 are constant to be determined therefore y = y_c + y_p = e^-2x (A cos x + B sin x) + c_1 sin x + c_2 cos x for same constant c_1 & c_2 \r\n R(x) = 2x^3 If is fundamental set is {1, x, x^2, x^3}