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.

 

[blue_raver22]

Sturm-Liouville form is a special form of differential equations that can be solved using techniques such as separation of variables and eigenfunction expansion. It is commonly used in mathematical physics and engineering to solve problems involving boundary value conditions.

In your case, it seems like you have a second-order differential equation with a coefficient function that is dependent on the variable x. This is not in the traditional Sturm-Liouville form where the coefficient function is constant. However, with some manipulation, it is possible to transform your equation into a Sturm-Liouville form.

One approach could be to use a substitution, such as u(x) = f(x)/x, to transform your equation into a form where the coefficient function is constant. This will require some algebraic manipulation and substitution of the derivative of u(x) into your equation.

Another approach could be to use a change of variable, such as y = x^2, to transform your equation into a form where the coefficient function is constant. This will also require some algebraic manipulation and substitution of the derivative of y into your equation.

Once you have transformed your equation into a Sturm-Liouville form, you can then use techniques such as separation of variables or eigenfunction expansion to solve for the function f(x) and normalize it.

It is important to note that there may be other techniques or methods that could be more suitable for your specific problem. It is always helpful to consult with a colleague or a textbook to explore different approaches and find the most appropriate method for your particular situation.

In summary, transforming your equation into a Sturm-Liouville form can be a useful approach to solving your problem and finding a normalized solution. I hope this helps and good luck with your research!