This is not a simple question. Solving d²y/dx² = - x² y
This is in the form of Fowlers' second order non linear differential equation.
Solution is not known in the form of a simple expression in terms of functions. So we will do using Taylor series expansion form of infinite series for y (x).
As the second derivative y" is a multiple of x² of y itself, we can find two independent different solutions one series with even powers or x and one with odd powers of x.
When we differentiate y wrt x twice, the power of x in series reduces by 2 in eq(2). But in equation (1), we have that y" has each term with power of x increased by two. Hence, in the equation (2), we must have powers of x of successive terms with a difference of 4. So then:
Now we use equation 1, to find the coefficients for the two independent solutions.
Compare equations (5) with (7) and (6) with (8) we get the coefficients:
Finally, any linear combination of the two above solutions will be the general solution the given differential equation in (1). y(x) = c₁ * y₁(x) + c₂ * y₂(x)