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# Power series solution

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Compute the power series solution for the ODE y''+x^2y=0 and determine the radii of convergence for the two series solutions.

https://brainmass.com/math/real-analysis/power-series-solution-224930

#### Solution Preview

The solution is in the attached file.

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------------------------------(i)

Comparing the equation with the equation
, we get

does not vanish at .
Then is an ordinary point.

Let the solution in series form be

---------------(ii)

------------(iii)

Putting the values of in (i), we get

or,

This is an identity. Equating the coefficients of various powers of to zero, we get the following relations:

------------------(iv)

------------------(v)

----------------(vi)

...

#### Solution Summary

This provides an example of power series solution and radii of convergence.

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