Cross multiplication gives

where we integrated. Now we take the exponential of both sides, i.e.

The last piece of the puzzle is to solve the equation involving the forcing term, where . In this case,  so we need to solve

The -16 comes from the expansion, where we had

The homogeneous equation is

With solution , C being the constant of integration. Since the inhomogeneous term is a constant, we guess that the particular solution is also a constant, call it A. So we have

Applying the initial condition we have This allows us to write

Now notice that

We can rewrite this as

Now,  is just . Comparison with the differential equation we started with for this term

tells us that it must be the case that , and so we take . To write down the final solution, we just write down a summation of the non-zero coefficient functions  multiplied by . And so we have

Notice it satisfies the initial condition  (set t = 0). What about long term behavior? This is called steady state behavior. We take the limit  and find that the steady state solution is