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I'm having a really hard time solving

$$xy'' + 2y' + 4xy = 0$$

I basically tried a lot of substitutions and series but couldn't find it. I realize this isn't as hard as I make it out to be so I'm assuming there is a certain trick that I am not seeing.

Thanks in advance!

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Have you seen $y''+p(x)y'+q(x)y=0$ before? – Jack Dec 11 '11 at 13:29
up vote 8 down vote accepted

Let be $z = xy$. We have $z''= 2y' + xy''$ e the equation becomes $$ z'' + 4 z = 0 $$

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Simply use the Taylor series. It will give you the right solution. Alternatively, you can try to reduce the 2nd order ODE to a system of 1st order ODEs. But anyway, the Taylor series seems to be an always viable approach.

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