First, find the characteristic solution. The characteristic equation for this ODE is
which has one root at
with multiplicity 2. This means the characteristic solution takes the form
There's no conflict with the nonhomogeneous part, which means you can guess a particular solution with undetermined coefficients of the form
which has derivatives
Substituting the particular solution into the ODE yields
Matching up coefficients gives a system of equations with solution
so that the particular solution is
which in turn means the general solution is
Use the initial conditions to solve for the remaining constants.
Therefore the solution to this IVP is