Writing Prompt(s)
We have seen that making a ``good guess'' can turn the problemof solving a second order linear differential equation into findingthe roots of a quadratic equation; specifically, for constantcoefficient equations we make the guess y = e m xand forCauchy-Euler equations we make the guess y = x m. Naturally, we'dlike to know if there are other types of second order linear DEsthat can be solved in this fashion. Are these the only ones or canyou find others? For example, is there a type of second orderlinear DE that can be solved by guessing y = sin â¡ ( m x ) andfinding the roots of an associated polynomial?
