Don't omit "stuff" from your post. It just happens that the "stuff" is easily identifiable, but try not to omit it.
This is an eigenvalue problem. Psi is just a function of x, it is your eigenfunction. H operating on Psi gives Psi multiplied by a constant. This constant is E.
As you asked not to do the math, the preceding information should suffice for you to find the solution.
If you're still having problems, ask yourself this question: what function would give itself multiplied by a constant as its second derivative?