Let

This has two "arbitrary" constants. I'll get to that in a minute.

Then

and

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

:

So

and

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:

such that

-Dan