Question

Solve the differential equation d2y
/dx2 + 2dy/dx + 4y = 111e2x cos3x
using the method of undermined coefficients.

Answer 1

The corresponding homogeneous differential equation

$$y''+2y'+4y=0$$

Characteristic equation

$$r^2+2r+4=0$$

$$r_1=-1-i\sqrt{3}, r_2=-1+i\sqrt{3}$$

The general solution of the homogeneous differential equation is

$$y_h=C_1e^{-x}\cos(\sqrt{3}x)+C_2e^{-x}\sin(\sqrt{3}x)$$

Find a particular solution of the nonhomogeneous differential equation

$$y_p=Ae^{2x}\cos(3x)+Be^{2x}\sin(3x)$$

$$y_p'=2Ae^{2x}\cos(3x)-3Ae^{2x}\sin(3x)$$

$$+2Be^{2x}\sin(3x)+3Be^{2x}\cos(3x)$$

$$y_p''=4Ae^{2x}\cos(3x)-12Ae^{2x}\sin(3x)$$

$$-9Ae^{2x}\cos(3x)+4Be^{2x}\sin(3x)$$

$$+12Be^{2x}\cos(3x)-9Be^{2x}\sin(3x)$$

Substitute

$$-5Ae^{2x}\cos(3x)-12Ae^{2x}\sin(3x)$$

$$+12Be^{2x}\cos(3x)-5Be^{2x}\sin(3x)$$

$$+4Ae^{2x}\cos(3x)-6Ae^{2x}\sin(3x)$$

$$+4Be^{2x}\sin(3x)+6Be^{2x}\cos(3x)$$

$$+4Ae^{2x}\cos(3x)+4Be^{2x}\sin(3x)$$

$$=111e^{2x}\cos(3x)$$

$$e^{2x}\cos(3x):3A+18B=111$$

$$e^{2x}\sin(3x):-18A+3B=0$$

$$A=1, B=6$$

$$y_p=e^{2x}\cos(3x)+6e^{2x}\sin(3x)$$

The general solution of the nonhomogeneous differential equation is

$$y=y_h+y_p$$

$$y=C_1e^{-x}\cos(\sqrt{3}x)+C_2e^{-x}\sin(\sqrt{3}x)$$

$$+e^{2x}\cos(3x)+6e^{2x}\sin(3x)$$