See also: http://brainly.in/question/183933. which I have solved.
This differential equations is called Fowler's second order differential (non-linear) equation. The solution is not found in closed expression of known simple functions. We can find the answer by using Taylor series expansion for y(x).
Solving d²y/dx² = - x² y
When we differentiate y(x) wrt x twice, the exponent of x reduces by 2 in y". On the RHS of (1) we have x² y, so exponent is increased by 2. Equation (1) will be valid if the exponent of x of (n+1)th term = 4 + exponent of n th term.
We can find two linearly independent solutions as:
Similarly for another independent solution :
Now we have the final general solution as :
here c1 and c2 are real constants.