Need Help Interpreting the Sturm-Liouville Operator

Question

I am given the following "Sturm-Liouville Problem with Operator $\mathcal{L}$ ": $$\mathcal{L}_{SL}=-\frac{1}{x}\left[\frac{d}{dx}\left(x\frac{d}{dx}\right)-\frac{1}{x}\right]$$

which is defined on some domain $D=\{x:x\in [0,1]\}$, and we were given some homogeneous Dirichlet Conditions of $x=0$ and $x=1$. We are asked to solve the problem with a Rayleigh-Ritz approximant method and compare to the exact answers obtained through solving it normally. The first 2 eigenfunctions and eigenvalues are required only. I am calling it $\mathcal{L}_{SL}$ to avoid confusion.

The operator confuses me and hence I cannot start solving the question. I would like to obtain a differential equation to work with.

I know that the S-L operator should take this form: $$\mathcal{L}[y]=\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y$$ Which is different to what we are given.

Attempt at understanding the question:

1. To get started, it seems that both sides of the operator $\mathcal{L}_{SL}$ needs to be multiplied by say $y$:

   $$y\mathcal{L}_{SL}=-y\frac{1}{x}\left[\frac{d}{dx}\left(x\frac{d}{dx}\right)-\frac{1}{x}\right]$$ This yields: $$\mathcal{L}_{SL}[y]=-\frac{1}{x}\left[\frac{dy}{dx}\left(x\frac{d y}{dx}\right)-\frac{y}{x}\right]=-y''-\frac{-y}{x^2} \Leftrightarrow y''-\frac{y}{x^2}=0$$

2. If we are to determine the quantities $p(x)$ and $q(x)$, we have:

   $$p(x)=-1$$ $$q(x)=\frac{1}{x^2}$$

3. The characteristic equation gives two distinct roots:

   $$\lambda=\pm \frac{1}{x}$$

4. To find the general solution, we were given Dirichlet Boundary Conditions which translates to $y(1)=y(0)=0$ which can be plugged into the general solution to determine the two required constants, or rather the eigenfunctions. This however feels odd. Up till now, I have been dealing with S-L BVPs which result in the form of either sines or cosines, and we use the fact that for example $\sin(n\pi)=0\forall n\in \mathcal{Z}^+$ to obtain the eigenfunctions. The roots have always been complex. In this case, they are not.

Am I heading in the right direction? Are my steps illustrated in the list 1~4 above correct? I have a feeling that I've misunderstood some fundamental basics to this type of problems. I do not know how to apply the Rayleigh-Ritz approximant and will look into it after I've solved the question for the two exact eigenvalues required.

Any help will be appreciated, thanks!

Answer 1

So, some fundamental errors here. First of all:

$$\frac{dy}{dx}\left(\frac{dy}{dx}x\right) \ne y''x$$

This mistake is just silly. Using the chain rule:

$$\frac{dy}{dx}\left(\frac{dy}{dx}x\right)=xy''+y'$$

We then multiply both sides of the differential operator by $y$, and replace $\mathcal{L}$ with $\lambda$:

$$0=y''+\frac{1}{x}y'+y\left(\lambda-\frac{1}{x^2}\right)$$

This should look oddly familiar, yes, it is in the form of a Bessel Differential Equation:

$$0=x^2y''+xy'+y(x^2-\alpha^2)$$

Getting to the general solution is a matter of playing with numbers:

$$y(x)=C_1J_1(\sqrt\lambda x)+C_2Y_1(\sqrt\lambda x)$$ Where $J$ and $Y$ are the Bessel Functions of the first and second kind.

Applying our Dirichlet Boundary Conditions, $C_2=0$ since:

$$\lim_{x \to 0}Y_1(\sqrt \lambda x)=\infty$$

The rest shouldn't be hard. Time to look up the approximant method.