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!


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:


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


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


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.


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