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

`(\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)}`