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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 --- (1)

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.

Solving d²y/dx² = - x² y --- (1)

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.