Its a Cauchy-Euler equation.
So suppose that the equation has a solution of the form . Plugging this back into the ODE yields the characteristic equation .
In this particular problem, has the characteristic equation , which yields repeated roots.
So once you have , you suppose . It turns out that once you write the ODE in the form and substitute into it, we see that (this is the reduction of order technique as seen in my tutorial here.)
Then at this point, we see that is the second solution.
Can you continue with the problem?