# Solve in series equation d2y/dx2+x2y=0

1
by TamraBansal305
It cnt b colv
It cnt be solves
Becz it hve only one

2015-09-13T21:37:42+05:30

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This is not a simple question.

Solving    d²y/dx² = - x² y    --- (1)

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).

---- (2)

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)

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