Second order ODE with non-constant coefficients

**toobusy** · October 21st 2011, 12:17 PM

I am trying to solve y''+(3-1/x^2)y=0. Is it possible to solve by separation? Or will this require a series solution?

**HallsofIvy** · October 21st 2011, 01:28 PM

If by "separation" you mean writing it as $\frac{d^2y}{dx^2}= (\frac{1}{x^2}- 3)y$ and then to
$\frac{d^2y}{y}= (\frac{1}{x^2}- 3)dx^2$ , then, no, you cannot "separate" a second order derivative like you can a first order derivative. The "differentials" dy and dx are defined in Calculus, but such things as " $d^2y$ " and " $dx^2$ " are NOT defined.

**Ackbeet** · October 21st 2011, 05:52 PM

Originally Posted by toobusy

I am trying to solve y''+(3-1/x^2)y=0. Is it possible to solve by separation? Or will this require a series solution?

It's a homogeneous linear equation, and thus always has the zero solution. However, if you desire non-trivial solutions, then series is the way to go.

**toobusy** · October 21st 2011, 07:39 PM

Thanks! I just figured it out!

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