Look at the picture below please
1 answer:
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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.
Substituting into the ODE yields
Now letting
Assuming
and integrating both sides with respect to
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),