Power series rep, and radius of convergence help!

Find all singular points and radius of convergence with power series

1) cos(2x) * y' + 4y = 12
2) (x^2+2)y''- (x/x+5)y = e^x
3) y''+sin(x)y=x^2

The nonhomogeneous part confuses me...

Assuming Maclaurin Series:

1) When cos(2x) = 0.
==> 2x = π/2 + πk for any integer k.
==> x = π/4 + πk/2 for any integer k.

So, the radius is π/4 (from the point of expansion x = 0 to the nearest singularity).

2) When x^2 + 2 = 0
==> x = ±i√2.

So, the radius is √2 (from the point of expansion x = 0 to either -i√2 or i√2).

3) No singular points, and hence infinite radius.

I hope this helps!

I was applying a theorem to allow you to anticpate the radius of convergence without solving for the series solutions.

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what about y''+(sin(x)/x)y=0? I know Sinx/x = 1

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