First find the characteristic solution. The characteristic equation is
which as one root at
of multiplicity 2. This means the characteristic solution for this ODE is
For the nonhomogeneous part, you can try a particular solution of the form
which has derivatives
Substituting into the ODE, the left hand side reduces significantly to
and it follows that
Therefore the particular solution is
and so the general solution is the sum of the characteristic and particular solutions,
which as one root at
For the nonhomogeneous part, you can try a particular solution of the form
which has derivatives
Substituting into the ODE, the left hand side reduces significantly to
and it follows that
Therefore the particular solution is
and so the general solution is the sum of the characteristic and particular solutions,