This has two "arbitrary" constants. I'll get to that in a minute.
Now, if we try to construct something like
we'll get exponential terms in the solution. But if we use:
we'll get polynomials.
Since we have no polynomial terms in the solution with specific (that is to say, constrained) coefficients, .
So input the solution :
Thus and which further gives , so the differential equation is
For simplicity we may drop the "a":
Now, how to get the form of to ? Well, this is a second order differential equation, so we need two constraint conditions. We are free to choose what conditions to specify, so long as the system is not overconstrained. But note that we have one undetermined variable, so we need only one condition. I would recommend something like:
It's a bizarre constraint, but it works.
So one possible differential equation would be: