Answer to Question #220881 in Differential Equations for Unknown346307

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

$y=y_h+y_p$

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

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