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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Sturm-Liouville solution satisfies some integral equation involving sine function

Let $u(s,\lambda)$ solves $$\frac{d}{ds} \left( p(s) \frac{du}{ds}(s) \right) - q(s)u(s) + \lambda u(s) = 0, \ s \geq 0$$ with boundary condition $$u(0,\lambda) = -\sin(h), \qquad p(0)u_s(0,\lambda) =...
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How to prove that the eigenvalues of this Sturm-Liouville problem are all positive?

I have been trying to solve the following problem: Let the Sturm-Liouville problem defined by $p(x)y''(x)+p'(x)y'(x)-q(x)y(x)+\lambda r(x)y(x)=0$ in the interval $[a,b]$, with the periodic ...
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General asymptotic result for $N(\lambda)$ in determining asymptotic distribution of eigenvalues

I am looking for the proof of The general result for $N(\lambda)$ for a Sturm-Liouville operator. $$\lim_{\lambda \to \infty} \frac{N(\lambda)}{\lambda^{d/2}} = \frac{v_d}{(2 \pi)^2}\int_{\Omega} ...
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Sturm-Liouville theory with Neumann boundary conditions

I am studying a problem concerning the properties of the eigenvalues and eigenfunctions of the following linear problem, $$ -u''+Vu=\sigma u\quad \text{in } (p,q), $$ $$ u'(p)=u'(q)=0, $$ where $[...
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Generalised orthogonality relations from Sturm-Liouville problems

I am familiar with the orthogonality relation for the Sturm-Liouville problem. Namely that the (normmalised) eigenfunctions of $$ \frac{d}{dx}\left(p(x)\frac{dy}{dx}\right)+q(x)y=-\lambda w(x)y $$ ...
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Seperation constant in PDE's

When should we assume the seperation constant in a seperation of variable Sturm-Liouville problem should be negative. I.e X''/X =-α
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How to solve $y''(t)+λy(t)=0 $

How to solve this Sturm-Liouville problem: $$y''(t)+λy(t)=0 $$ $$y(0)=y'(\pi) ,y'(0)=y(\pi)$$ Would really appreciate a solution or a significant hint because I couldn't find anything
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Rayleigh Quotient on Legendre´s Differential Equation

We know that Legendre's DE $$ \frac{d}{d\phi}\bigg(sin\phi\frac{dg}{d\phi}\bigg) + \big(\mu - \frac{m^2}{sin\phi}\big)g = 0 $$ can be transformed by letting $x = cos\phi$ into $$ \frac{d}{...
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Solving $y''=\lambda y$

"Show that all solutions of $y''(x)=\lambda y(x)$ on $0\leqslant x \leqslant L$ with $y(0)=y(L)=0$ are of the form $c\sin\left(\frac{k\pi}{L}\right)x.$ (Hint: write down all solutions of the o.d.e and ...
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How to solve a Sturm-Louiville Problem with nonhomogeneous BCs

I'm trying to solve the following steady-state problem: $$\nabla^2 u = 0 , \ \ \ (x,y,z) \in [0,\ 2]^3$$ Insulated in the faces with $x=0,y=2$. Kept at $0º$C on the faces with $x=2,z=0$. And ...
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How do you solve the following nonhomogeneous eigenvalue problem?

Let's say we want to solve this Sturm-Louiville problem: $$ h^{"}(z) + \lambda h(z) = 0$$ subject to boundary conditions $$ h(0) = 0 \ ; h(5) = 25. $$ This is my approach: $\lambda = 0$ $$h(z) ...
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Constructing a Green function for Sturm - Liouville operator

I can't find Green function for the Sturm - Liouville operator : $$ L = - \frac{d²}{dx²} + 1 $$ with condition : $v(0) = v(1) = 0$ I would appreciate any help Thanks in advance
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roots of a solution of second order ODE [closed]

How can I prove that if a non trivial solution of the ODE $$u''+(a+b\cos 2x)u=0$$ that has $2n$ roots in the interval $(-0.5 \pi,0.5 \pi)$, then $$(2n-1)^2 \le a+b~?$$ thx in advance
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Sturm-Liouville system

Let $A$ be $n \times n$ real matrix and consider the following Sturm-Liouville system $$y''(x)=Ay(x) \\ y(0)=y(1)=0,$$ where 0 is the null vector in $\mathbb{R}^n$. How can we solve this system ...
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Sturm-Liouville eigenfunctions in a double Robin condition

I'm solving a PDE by using the method of separation of variables, and this led me to a Sturm-Liouville of the form $$ \dfrac{d^2f}{dx^2} + \lambda f = 0 \\[15pt] \left\{\begin{array}{l} \dfrac{df}{dx}(...
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finding number of positive and negative eigenvalues of a Sturm-Liouville problem

Consider the eigenvalue problem $$((1+x^4)y')'+\lambda y=0 \ , \ x\in(0,1) \\ y(0)=0 \ , \ y(1)+2y'(1)=0$$ Then which of the following are true? $(a)$ All the eigenvalues are negative. $(b)...
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There cannot be two linearly independent eigenfunctions in a Sturm–Liouville problem

Consider the Sturm–Liouville problem $y'' + [λp(x) − q(x)]y = 0$, $α_1y(a) + α_2y'(a) = 0$, $β_1y(b) + β_2y'(b) = 0$, with $p(x)$ and $q(x)$ continuous, $p(x) > 0$ on $[a, b]$, and $α_1β_1 > ...
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Is this equation exactly solvable?

I would like to know, if the following equation (which resembles the Sturm-Liouville equation) is exactly solvable (and how, if yes): $$\frac{d}{dx}\left[p(x)\frac{d}{dx}y(x)\right] - k^2p(x)y(x) = 0.$...
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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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Modified Sturms Comparison Theorem

Suppose that we have some smooth $x(t), y(t)$ satisfying $$ x''+Px=0, \; \; y''+Qy=0 $$ with $Q(t)\geq P(t)$ for all $t$ and $x(0)=y(0)=0$, $x'(0)=y'(0)>0$. I would like to show that $x(t)\geq y(t)...
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Common eigenvalues for two Sturm-Liouville problems

Does exist in literature any results concerning the common eigenvalues for the two eigenvalue problems of the form $$y''(x)=\lambda^2 y(x)+\lambda a(x) y(x), \ x\in(0,1), $$$$z''(x)=\lambda^2 z(x)-\...
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Doubt regarding Eigen values and Eigen functions of a boundary value problem.

Consider the differential equation ${{d^2y}\over{dx^2}}+\lambda y=0,\lambda>0$ where $y(-l)=y(l)=0,l>0$. Now from the two boundary conditions I am getting $C_1\cos(\sqrt{\lambda}x)-C_2\sin(\sqrt{...
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Sturm-Liouville Problem: Finding eigenvalues and eigenfunctions

I am trying to find the eigenvalues and eigenfunctions of the following Sturm-Liouville problem: $$ \begin{cases} -u''=\lambda u, \ &x\in (a, b), \\ u(a)=u(b)=0, \ &b>a.\\ \end{...
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Discreteness of spectrum of operator $(p(x)y'(x))' + q(x)y(x)$ with periodic coefficients

Suppose $p, q : \mathbb{R} \to \mathbb{R}$ are piecewise continuous and periodic of period $a > 0$. Moreover, suppose $p$ is nowhere zero and $p'$ is piecewise continuous. I would like to ...
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Solving another non-trivial recurrence relation

This is a generalization of question Yet another non-trivial recurrence relation to solve. . Let $E \in {\mathbb R}$ and $\lambda^{C} \ge 0$, $\lambda^{M} \ge 0$ and $\Lambda \ge 0$.In addition to ...
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Put each equation in self-adjoint form (Sturm-Liouvulle)

Put each equation in self -adjoint form $x^{2}{y}''-x{y}'+\lambda y=0$ If I want to put it in the self-attached form of Sturm Liouvulle, I first do this: $x{y}''-{y}'+\frac{\lambda }{x}y=0$ $e^{-\...
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Rayleigh quotient on circular region of radius 2

I' m struggling with the following problem. We have the eigenvalue problem: $$u'' + \lambda u = 0$$ with associated boundary condition: $$u' + 3u = 0$$ Now by using the Rayleigh quotient for $0 \leq ...
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Sturm-Liouville exercise: finding linearly independent solutions

Consider the problem $Lu:=u''(x)+{1\over 4x^2}u(x)=0,\ u(1)=u(2)=0$ Green's function associated with $L$ is defined as $\Gamma(x,\xi)=\begin{cases}{1\over c}u_1(\xi)u_2(x),\ x\ge\xi,\ x\in[1,2]\\ {1\...
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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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Show that the function is orthogonal

show directly That the eigenfunction of the given Sturm-Liouville problem are orthogonal without explicitly solving for therm , and state the orthogonality conditions $x{y}''+{y}'+\lambda xy=0$ ${y}'...
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show that $λ=0$ is an eigenvalue of the regular Sturm- Liouville system

show that $λ=0$ is an eigenvalue of the regular Sturm- Liouville system $\frac{\mathrm{d} }{\mathrm{d} x}[p(x){y}']+\lambda r(x)y=0$ ${y}'(0)=0$ ${y}'(1)=0$ What I have done is this: $\frac{\...
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Verify that the periodic Sturm- Liouville system has a symmetric operator

Verify that the periodic Sturm- Liouville system has a symmetric operator The ones that I know is that the Sturm-Liouville periodic system is: $L[y]=\lambda r(x)y=0$ $x_{1}< x< x_{2}$ $y(...
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show directly that the eigenfunction of the given Sturm-Liouville problem are orthogonal

show directly That the eigenfunction of the given Sturm-Liouville problem are orthogonal without explicitly solving for therm , and state the orthogonality conditions ${y}''+\lambda (1+x)y=0$ $y(0)=...
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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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By changing the variable in the φ equation to x = cos φ, derive the self adjoint form of the Legendre equation

We have $$\frac{d}{d\phi}(sin\phi \frac{dP}{d\phi}) + \lambda sin\phi P =0 $$ By changing the variable in the equation to x = cos $\phi$, derive the self adjoint form of the Legendre equation: $$\...
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Solving Laplace's Equation on the Semi-annular Region (With Solution)

I'm solving this problem part (b) and I am stuck on the last part of the question. The solution is posted here. Where I don't get is this part: I am assuming that the professor is trying to ...
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Show that eigenvalues are non-negative from the two-dimensional Rayleigh Quotient

For the eigenvalue problem: \begin{align*} \nabla^2 \phi + \lambda \phi &= 0 \\ \end{align*} with the boundary condition along the entire boundary: \begin{align*} a(x,y) \phi + b(x,y) \...
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3answers
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Determining Weight function in Sturm Liouville problem

By choosing the proper weight function $\sigma (x) $ solve the Sturm-Liouville problem and determine its eigenvalues and eigenfunctions. $$ \frac{d}{dx}\left[x\frac{dy(x)}{dx}\right] + \frac{2}{x}y(x)...
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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 ...
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Infinite amount of negative eigenvalues for SL problem when p < 0?

In this chapter from a book, people.cs.uchicago.edu/~lebovitz/Eodesbook/sl.pdf, on p.201, a theorem is stated and later proven regarding the existence of an infinite set of negative eigenvalues of a ...
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solution of Given boundary value problem

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 ...

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