It's a nonhomogeneous linear equation. So you'll need to find a particular solution of the nonhomogeneous equation and the general solution of the related homogeneous equation
The homogeneous equation is
The characteristic equation is , which has roots of r=6 and r =1.
So the general solution of the homogeneous equation is
Now we need any particular solution of
so the general solution on the nonhomongeneous equation is
Now we need to find the two coefficients by using the initial conditions.
Solving the two equations simultaneously, and
so our final solution is