Hmm am I missing something obvious but how would I solve y"=ay^2?
For this type of second order ODE the 'standard approach' is based on the identity...
(1)
... so that the equation you have prosed becomes...
(2)
... that is 'separated variables' and the solution of which is...
(3)
Also (3) is 'sepatated variables' and its solution is...
(4)
... even if the integral in (4) is not very 'confortable'...
Merry Christmas from Italy