Question

a mechanical system is governed by the following differential equation what is the homogeneous solution
d^2y/dt^2 + 6 dy/dt + 9y = 4e^- t

Answer 1

The ODE has characteristic equation

$r^2+6r+9=(r+3)^2=0$

with roots $r=-3$, and hence the characteristic solution

$y_c=C_1e^{-3t}+C_2te^{-3t}$

For the particular solution, assume an ansatz of $y_p=ae^{-t}$, with derivatives

$\dfrac{\mathrm dy_p}{\mathrm dt}=-ae^{-t}$

$\dfrac{\mathrm d^2y_p}{\mathrm dt^2}=ae^{-t}$

Substituting these into the ODE gives

$ae^{-t}-6ae^{-t}+9ae^{-t}=4ae^{-t}=4e^{-t}\implies a=1$

so that the particular solution is

$\boxed{y(t)=C_1e^{-3t}+C_2te^{-3t}+e^{-t}}$