How Can I Transform My O.D.E. into Sturm-Liouville Form?

[kalphakomega]

Close to Sturm-Liouville form...

I got an O.D.E down to the form

f^ll(x) + ($$\lambda$$ - 16x²)f(x) = 0

I omitted some constants to make it look simple. What I'm trying to do is find a function f(x) to normalize. Solving by using roots ended up giving me an exponential function I am unable to solve. However I think if I could convert the above to proper Sturm-Liouville form I might find an alternative expression for y(x) so that I could normalize its square. Any thoughts? I'm not completely competent in the aspects D.E as of yet so I may have missed a simpler route. Input is greatly appreciated.

 

[yungman]

You look into modified Bessel, Parametric Bessel equation? I don't have the answer, just look very similar to one of those.

 

[gato_]

I don't know what you mean by using roots, but your equation is inhomogeneous. This is actually a rescaled version of the quantum harmonic oscillator, the solutions of which are given in terms of Hermite functions. Look here for a concise reference:
http://www.fisica.net/quantica/quantum_harmonic_oscillator_lecture.pdf

 

[kosovtsov]

The general solution to your ODE is as follows

$$f(x) = \frac{1}{\sqrt{x}}[C_1 WhittakerM(\frac{\lambda}{16},\frac{1}{4},4x^2)+C_2 WhittakerW(\frac{\lambda}{16},\frac{1}{4},4x^2)]$$

where C₁ and C₂ are arbitrary constants.