# Questions tagged [sturm-liouville]

The Sturm–Liouville equation is a particular second-order linear differential equation with boundary conditions that often occurs in the study of linear, separable partial differential equations.

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### SL-Eigenvalue/function problem with arbitrary boundary values

The problem is to find all eigenvalues and eigenfunctions for the following SL system. $u'' + \lambda u = 0, x \in [a,b]$ $u'(a) = u'(b) = 0$ I know the general idea of how to do these problems ...
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### The Node Theorem - an argument from physics

In the mathematics literature, the theorem falls under the Oscillation Theory of the Sturm-Liouville equation. It doesn't seem to have a special name in the mathematics literature, but it is well-...
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### When do singular Sturm-Liouville problems admit the oscillation property (like e.g. Hermite Polynomials do)?

I refer to Sturm-Liouville problems as \begin{equation} \tag{1} \frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y \end{equation} The ...
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### Sturm-Liouville problem with approximated coefficients

Consider the problem $$f(x)y(x)''+\lambda y(x)=0$$ with $$y(0)=y(1)=0$$ In many cases it is not immediate finding the analytical solution of the differential equations because it depends on the ...
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### How to show singular Sturm-Liouville problem has no eigenvalues?

I have the following SL problem: $$(x^{2} f')'+ \lambda f = 0$$ where $\lvert f(x) \rvert$ is bounded as $x \rightarrow 0$ and $f(1) = 0$ I have to show that the above problem has NO ...
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### example of Sturm-Liouville problem with Lu=u'' differential operator

I'm in the process of understanding Sturm-Liouville theory and I was reading this example Sturm-Liouville example from wikipedia. I don't understand why they solve $Lu=x$, like where does this come ...
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### Solving PDE with interesting initial condition

I have the PDE (Fokker-Planck) which reads $$\frac{\partial p}{\partial L}(L,x) = \bigg((x-1)^2\frac{\partial^2p}{\partial x^2}(L,x)+4x(x-1)\frac{\partial p}{\partial x}(L,x)+2xp(L,x)\bigg),$$ with ...
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### Let $P_n(x)$, $n=0,1,2,…$ be an infinite set of polynomials satisfying the three conditions

Let $P_n(x)$, $n=0,1,2,...$ be an infinite set of polynomials satisfying the three conditions $P_n(x)$ is of degree $n$ $P_n(1)=1$ ,$n=0,1,2,...$ $\int_{-1}^{1}p_{n}(x)p_{k}(x)dx=0$, $n\neq k$ ...
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### Find the expansion of $f (x) = x, 0 ≤ x ≤ \pi$ in a series of eigenfunctions of the Sturm–Liouville system $y''+\lambda y = 0, y (0) = 0, y (\pi)=0.$

I need to expand the given function in terms of eigen functions series, firstly i find the solutions to the SLP and from there i get my eigen functions $n$ values, and then i find the coefficient Cn ...
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### How do I apply sturm comparison theorem to draw the conclusion?

Let $y$ be a non-trivial solution of the boundary value problem, $$y"+xy=0;x\in [a,b];y(a)=y(b)=0.$$ Then $b>0$ $y$ is monotone in $(a,b)$ if $a<0<b$ $y'(a)=0$ $y$ has infinitely many zeroes....
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### Expansion of $ln(\operatorname{cosec} {x})$ in terms of Spherical harmonics or Legendre Function

I need the coefficients in: $$ln(\operatorname{cosec} {x})= \sum_{0}^{\infty} \sum_{-l}^{l} A_{lm} \mathcal{Y(\theta,\phi)}$$ So, I used here standard technique to multiply both sides by the ...
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### Find eigenvalues of $x^2 \frac{d^2\phi}{dx^2} + x \frac{d\phi}{dx} + \lambda \phi = 0$ with boundary conditions $\phi(1) = \phi(b) = 0$

Since this is an equidimensional equation, determine all positive eigenvalues. \begin{align*} x^2 \frac{d^2\phi}{dx^2} + x \frac{d\phi}{dx} + \lambda \phi &= 0 \\ \phi(1) &= 0 \\ \phi(...
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### Sturm-Liouville terminology clarification

I need some clarification about what exactly the 'Sturm-Liouville form' is. The 'Sturm-Liouville problem' arises when, given $$a(x)y''(x)+b(x)y'(x)+c(x)y = f(x): a(x), f(x) \neq0$$ I rewrite the ...
For $\lambda\in \mathbb{R}$ consider the boundary value problem $$x^2 \frac{d^2y}{dx^2}+2x\frac{dy}{dx}+\lambda y=0\;\;\;,y(1)=y(2)=0$$ called as $(P_{\lambda})$ Then i Have to prove which of ...