The general solution to the differential equation I am having problems in classifying the differential equation $y''=y(x^2)$ in categories like homogeneous, exact, bernoulli, separable and non-exact so I could have the general solution. 
Or would someone help me find the solution 

The General solution to this differential equation would be an infinite series:
$$y(x) = \sum^\infty_{n=0} a_nx^n $$
In this case, the coefficents are pretty odd.  Since $ a_2 = 0, a_3 = 0 $, and the recurrence is every 4 terms.  In this case we have:
$$ a_4 = a_0/12, a_5 = a_1/20, a_8 = a_4/56, a_9 = a_5/72 $$
To find the exact relationship requires some art.  I would go with:
$$ a_{4k} = \frac{a_0}{(4^k)k!(4k-1)(4k-5)...etc)} , a_{4k+1} = \frac{a_1}{4^kk!(4k+1)(4k-3)...etc} $$
The etc. can be rewritten as factorials if needed.
