# Solve the series the eqution d2y/dx2+x2y=0

1
by TivriRout

2015-09-20T00:18:13+05:30

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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.

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