Question

x²y''-7xy'+16y=0 find general solution​

Answer 1

Answer:

Given a solution , we can attempt to find another via reduction of order of the form . This has derivatives

Substituting into the ODE yields

Now letting , so that , you end up with the ODE linear in

Assuming (which is reasonable, since is a singular point), you can divide through by and end up with

and integrating both sides with respect to gives

Back-substitute to solve for :

and again to solve for :

Alternatively, you can solve this ODE from scratch by employing the Euler substitution (which works because this equation is of the Cauchy-Euler type), . You'll arrive at the same solution, but it doesn't hurt to know there's more than one way to solve this.