Question

Solve d^{2}y/dx^{2}-6dy/dx+13y=20e^{3x}cos(1x+5)

Answer 1

Given DE:

$$y''(x)-6y'(x)+13y(x)=20 e^{3x}\cos (x+5)$$

The characteristic equation for homogeneous DE

$$k^2-6k+13=0$$

Roots:

$$k_1=3-2i,\quad k_2=3+2i$$

Thus, the general solution of homogeneous DE

$$y_h(x)=e^{3x}(C_1 \cos 2x+C_2\sin 2x)$$

The partial solution of inhomogeneous DE

$$y_i(x)=\frac{20}{3}e^{3x}\cos(x+5)$$

Finally,

$$y(x)=y_h(x)+y_i(x)=\\ =e^{3x}(C_1 \cos 2x+C_2\sin 2x)+\frac{20}{3}e^{3x}\cos(x+5)$$