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

Question

asked Jul 15, 2021 31.2k views

1 Answer

Kkuilla · Answer 1 · 2021-07-20T23:59:59+0000

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

$(\mathrm dy_p)/(\mathrm dt)=-ae^(-t)$

$(\mathrm d^2y_p)/(\mathrm dt^2)=ae^(-t)$

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

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