Question

Writing Prompt(s)

We have seen that making a ``good guess'' can turn the problem of solving a second order linear differential equation into finding the roots of a quadratic equation; specifically, for constant coefficient equations we make the guess y = e m xand for Cauchy-Euler equations we make the guess y = x m. Naturally, we'd like to know if there are other types of second order linear DEs that can be solved in this fashion. Are these the only ones or can you find others? For example, is there a type of second order linear DE that can be solved by guessing y = sin ⁡ ( m x ) and finding the roots of an associated polynomial?

Answer 1

Yes, there are other types of second order differential equation that can be solved by guessing the solution like you said.

1) For Shrodinger equation of a particle in a Infinite box potential (inside the box potential V = 0), wave function of the particle can be solved by guessing y = sin (m x).

2)

3)