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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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Eigenfunction expansion to solve non homogenous heat equation

I've been really struggling to figure out how to solve this problem using Eigenfunction expansion, I can solve it using seperation of variables. So this the problem is: $$ \begin{cases} u_t(x,t)=u_{...
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How to find Sturm-Liouville problem eigenvalue and function?

So I have the following Sturm-Liouville problem: $$ y'' + \lambda y = 0 $$ Such that $ \lambda > 0 $ and the initial conditions are as follows: $$ y (0) + y'(0) = 0 $$ $$ y(1) + y'(1) = 0 ...
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Find the eigenvalue and the eigenfunction of $(xy' )'+(1+λx)y=0$ , $-1≤x≤1, y(-1)=y(1)=0$

I believe the title explain what I want. It is Sturm Liouville problem and I tried to solve it by multiplying the whole equation with $1/x$ , so that the characteristic polynomial will be $r^2+λ+1/x=...
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Sturm-Liouville form question urgent

Hi guys so I have this differential of order 2 that I want to get to the Sturm-Liouville form by first finding the $p(x)$. The form itself is : $(p(x).y'(x))'+q(x).y(x)=0$ And of course, it has ...
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Show that the eigenvalues of a Sturm-Liouville problem are in a certain interval

Considering the Sturm-Liouville problem $$-y'' = \lambda y, $$ with $y(0)=0$ and $y'(1)-y(1)=0$, and where we let $\lambda_0 < \lambda_1 < \lambda_2 <...$ be the eigenvalues in increasing ...
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Show that any nontrivial solution to $-u'' + q(x)u = 0$ has finitely many zeros

I am looking through an example and need some help to understand it. For the problem $-u'' + q(x)u = 0$, where $0<x<1$, $q \in \mathcal{C}(0,1]$ and $x^2q(x) \rightarrow q_0$ as $x \rightarrow ...
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Sturm-Liouville, find normalized eigenfunction

I want to find the eigenvalues and normalized eigenfunctions of the problem $$-y'' = \lambda y, y'(0) = y(1) = 0. $$ By solving $r^2 + \lambda = 0$ I found the general solution $y(x) = c_1\cos(\sqrt{\...
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Proving that the eigenvalues of the Airy problem are positive

I am solving an exercise concerning the Airy eigenvalue problem $$ -y''+xy =\lambda x, \quad y(0)=y(1)=0, \quad (*) $$ which (among other things) asks me to prove that all eigenvalues are positive. I ...
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Differentiating Eigenvalues of a Schrodinger operator

Consider a one-dimensional Schrodinger operator of the form $$ H_h=-\frac{d^2}{dx^2}+V_h(x), \quad h>0, $$ on $L^2([-1,1])$. Let us further assume that the mapping $(0,\infty)\times[-1,1]\ni(h,x)\...
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Is this Sturm-Liouville problem self-adjoint?

We are interested in determining whether the problem $\begin{cases}xu''-u'+u = \cos(x)\\u(0) = 0 \\ u(1) = u'(1)\end{cases}$ is self-adjoint. This is not a Sturm-Liouville problem, the corresponding ...
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Wave PDE with Neumann and Dirichlet initial condition

So I'm given the following PDE problem, which I try to solve using variable separation: $\dfrac{d^2u(x,t)}{dt^2}-c^2\dfrac{d^2u(x,t)}{dx^2}=0$ $0<x<L,\enspace t>0$ $u(0,t)=0, \dfrac{du(L,t)...
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How to find the exact solution of a Sturm Liouville form, 2nd order ODE?

I have a second order ordinary differential equation reducible to Sturm Liovelle form. The equation is given by $\frac{f(x)}{x^2} - (\frac{1}{x}+x)f'(x) + f''(x) =0$ and boundary conditions are : $...
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Can this problem be reduced to a Sturm-Liouville form?

From a system of three coupled PDEs \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c ...
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Is 2nd-order linear ODE with quadratic coefficients solvable?

Consider a general 2nd-order ODE $$\alpha_2 y''(x) + \alpha_1 y'(x) + \alpha_0 y(x)=0$$ where coefficients are all real quadratic polynomials of $x$ $$\alpha_n=\sum_{i=0}^{2}{a_i^{(n)} x^i}.$$ Is it ...
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Is 2nd-order ODE with quadratic coefficients solvable?

Consider an ODE eigensystem $$\begin{bmatrix} 0 & d_1-\mathrm id_2 \\ d_1+\mathrm id_2 & 0 \end{bmatrix} \begin{bmatrix} a(y) \\ b(y) \end{bmatrix} = \lambda \begin{bmatrix} a(y) \\ b(y) ...
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Are all fourth-order self-adjoint differential operators also Sturm Liouville?

Suppose we have a fourth-order differential operator $L$ that we know is self-adjoint. Suppose also that the typical Sturmian boundary conditions are satisfied. Is $L$ also Sturm Liouville? (This is ...
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Possible spectra of singular Sturm-Liouville problems

A Sturm–Liouville (SL) eigenvalue problem with separated boundary condition on $[a,b]$ $$(py')'-qy=-\lambda^2wy$$ is regular if $p(x),w(x)>0$ and $p(x),p'(x),q(x),w(x)$ are continuous in the finite ...
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Fourth-order Sturm Liouville form

A fourth-order ODE in the form \begin{align} [p(x)f''(x)]'' - [q(x)f'(x)]' + s(x) f(x) = \lambda r(x) f(x), \end{align} is in a fourth-order Sturm Liouville form. In the simpler second-order problem, ...
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Sturm - Liouville Parseval proof

Consider the expansion $𝑓(𝑥)=\sum_{n=0}^{\infty}A_𝑛\phi_𝑛(𝑥)$ where $\phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable ...
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Limit circle/point of ODE with finite eigenvalues

Consider the following Sturm–Liouville (SL) eigenvalue problem $$(x^2y')'-[(x/2+a)^2+a]y=-\lambda^2y(x),$$ defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ and parameter $a>0$. ...
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Convert in a Sturm-Liouville form $y''+R(x)y'+(Q(x)+\lambda p(x))y=0$

Convert in a Sturm-Liouville form $$y''+R(x)y'+(Q(x)+\lambda P(x))y=0\tag1$$ My attempt: The form of Sturm-Liouville is: $$\frac{d}{dx}[r(x) \frac{dy}{dx}]+(q(x)+\lambda p(x))y=0\tag2$$ For obtain ...
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Determine the normalised eigenfunctions for the BVP: $y''+λy=0, y(0)=0, y(1)=0$

Solving it I get: $y(x)=c_1 \cos(x \sqrt{\lambda}) + c_2 \sin (x \sqrt{\lambda})$ $y(0)= C1 + 0 = 0, C1=0$ $y(1)=0+C2\sin(\sqrt{\lambda})=0$ So, $(\sqrt{\lambda})=n\pi$, $({\lambda})=(n\pi)^2$ So, ...
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How to find the Green's function for a non-homogeneous PDE?

Very general question on how to find the Green's function that satisfies $$y(x) = \int_a^b G(x,s)f(s)ds$$ For a non-homogeneous PDE problem of the form: $$L[y(x)] = f(x)$$ In the domain $[a,b]$. ...
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Sturm-Liouville eigenvalues have lower bound

I am looking to find a proof that eigenvalues are bounded below for the the general Sturm Liouville equation on an open region $ \Omega \subset \mathfrak R^d $ (perhaps with compact closure), $Ly -\...
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Analytical solution to unidimensional transient heat equation with mixed bounday conditions not zero

I need to solve the equation: ${\partial^2 T(x,t) \over \partial x^2 }= {1 \over \alpha} {\partial T(x,t) \over \partial t}$ With the boundary conditions $T(0,t)=T_0$ and ${\partial T(L,t) \over \...
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Exercise about the Sturm-Liouville problems

Under what condition on the constants $c$ and $c'$ are the boundary conditions $$f(b)=cf(a)$$ and $$f'(b)=c'f'(a)$$ self-adjoint for the operator the operator $$\mathcal{L}f=\frac{d}{dx}\left(p_0(x)\...
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1answer
51 views

Solve a Sturm-Liouville Boundary Value Problem

Solve the Wave Equation I've been trying to solve the above wave equation where $u = u(x, t)$ and $c ∈ \mathbb{R}$ is a constant, subject to $$ u(x, 0) = 0,\;\; 0 < x < 1, \\ u_t(x, 0) = U_0x,\...
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Sturm-Liouville Completeness Proof

I am looking for a basic but rigorous introduction to Sturm-Liouville theory. In particular, I would like to see a proof that the eigenfunction solutions of Sturm-Liouville problems are complete. ...
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Legendre's Equation, sturm liouville - eigenvalues/eigenfunction

Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction Consider the linear differential operator: $$ L = \frac{1}{4}(1+x^2)\frac{d^2}{dx^2}+\frac{1}{2}x(1+x^...
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Do ground states of the higher-dimensional Schrodinger equation admit nodes?

Neglecting some irrelevant physical constants, the one-dimensional Schrodinger equation for a single particle is the eigenvalue equation for a particular second-order linear ordinary differential ...
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1answer
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Cannot solve recursion relation in power series solution to this ODE

I'm trying to solve the differential equation $$\frac{d^2u}{dr^2} - \left[V_0(r-1)^2 + \frac{\ell(\ell+1)}{r^2} \right]u = -\lambda u$$ where $r\geq0$ is the radial component in spherical coordinates, ...
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1answer
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Sturm-Liouville Form (e.g. Bessel Equation)

I am trying to convert the Bessel equation $$z^2u''+zu'+(k^2z^2-v^2)u=0,$$ into the Sturm-Liouville form. Dividing by $z$ ($z\neq 0$), we yield $$zu''+u'+(k^2z-v^2z^{-1})u=0\implies (zu')'+(k^2z-v^2z^...
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Mode shapes and frequencies of Sturm-Liouville problem [closed]

I have been asked the following problem in my homework. Given a differential equation $$ xy^{''}+y^{'}+k^{2} y(x)=0$$ with condition $y(1)=0$ and $y(x)$ is finite when $x \to 0$ find mode shapes ...
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1answer
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Verifying an eigenvalue of a Sturm-Liouville problem

Consider the Sturm-Liouville problem $$(xy')'+\lambda y=0, 1<x<2$$ with boundary conditions $y(1)=y(2)=0$. The question of interest is whether $\lambda=0$ is an eigenvalue of the above SL ...
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2answers
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What is the error I am making on getting general solutions for this Sturm-Liouville problem?

Given this Sturm-Liouville problem: $$X'' + \lambda X = 0$$ There are general solutions (Eigenfunctions) for three cases on $\lambda$: $$\lambda > 0$$ Has the characteristic equation: $r^2+\lambda ...
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1answer
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Solution of equation $y''+x^2y=0$

Does equation $y''+x^2y=0,$ where $y$ is function of $x$ have explicit solution? Perhaps with some conditions or in special case? I came across that when we have $x$ instead of $x^2,$ solutions are ...
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Physical Intuition Behind Equidimensional Equation

Consider a string with a density profile, $\mu(x)$, given by: $$\mu(x)=\frac{\mu_0}{x^2},$$ assume a time dependence of $\sim e^{i\omega t}$, hence, the wave equation is given by: $$x^2\frac{d^2y}{dx^...
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2answers
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Don't understand this Sturm - Liouville problem

All the SL problems I have seen before had $ \lambda$ in them, so finding the eigenvalues meant finding the values of $ \lambda$. However I don't know what I am supposed to do in the following: Find ...
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2answers
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Solution of Sturm-Liouville problem

Here I have been asked one question in my homework if someone can give a hint, I would be grateful for it. If a function $f(x)$ has an expansion $$f(x)=\alpha_0 x+\sum_{n=1}^{\infty} \alpha_{n} \...
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Second order differential equation with complex initial values

I'm trying to solve the following second order differential equation (which is supposed to be solvable analytically): $\ddot y + 3\alpha \dot y +\beta^2 \exp\left(-2\alpha t\right) y=0$, being $\...
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1answer
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Need Help Interpreting the Sturm-Liouville Operator

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 ...
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Exercise about the Sturm-Liouville problem

I don't understand the following question about a Sturm-Liouville problem. Let $\{ f_n(x) \}$ be a family of functions that are mutually orthogonal from 0 to $\infty$ with function weight $e^{-x}$. ...
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Eigenvalues of the differential operator $-\frac{d^2}{dx^2} + k\left( \frac{x}{a} - \frac{a}{x}\right)^2$ on $(0,\infty)$.

I'm trying to solve for the eigenvalues of the differential operator $$-\frac{d^2}{dx^2} + k\left( \frac{x}{a} - \frac{a}{x}\right)^2$$ over square-integrable functions on $(0,\infty)$ where $k,a$ are ...
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1answer
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Solving an ODE that is piecewise defined using the Dirac Delta function

Fix $\xi \in (0,L)$. Find function $g(\cdot; \xi) : (0,L) \rightarrow \mathbb{R}$ satisfying $$ \begin{cases} -\displaystyle \frac{\rm d}{{\rm d} x}\Big( a(x) \frac{{\rm d} g}{{\rm d} x} \Big) = \...
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Is the solutions of Sturm-Liouville boundary problem still true for complex boundary?

Given the Sturm-Liouville problem on $0 < x < 1$: $$y" + \lambda^2 y = 0$$ Satisfying boundary conditions: $$y(0) - h_1y'(0) = 0$$ $$y(1) - h_2y'(1) = 0$$ In the general case the solution for ...
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63 views

what will be the eigenvalues and eigen functions for these equations?

I solved my PDE $\partial_X\theta - \frac{2}{Y(1-Y)}\partial_{YY}\theta = 0$ by using separation of variables (cf. this post). I got two solutions $F(X)$ and $G(Y)$ where $\theta(X,Y) = F(X)\, G(Y)$. ...
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1answer
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Sturm-Liouville Problem: Eigenvalues and general solution, defined on $|x|\le d, a<\pi /d$

I have a Sturm-Liouville BVP that I would like to evaluate, I want the general solution for eigenvalues, and I would like to plot maybe 10 of them: $$-y''+a^2y-b y=0 \in D$$ $$y'=0 \in \partial D$...
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Can I consider these equations as Sturm-Liouville problem?

I have two ODE's and I want to solve it but I am not confident that can I consider it to Sturm-Liouville type? if I can then what will be the general solution for these equations. \begin{aligned} F'(X)...
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1answer
78 views

Straightforward derivation leads to a more general Sturm-Liouville operator than expected

In this question, I want to investigate the detailed derivation of the Strum-Liouville self-adjoint operator $$Lu = - \frac{1}{w(x)} \left\{ \frac{d}{dx} \left[p(x) \frac{d u}{dx} \right] + q(x) u\...
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2answers
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Finding Solutions of Sturm-Liouville Equation Satisfying Boundary Conditions and Checking Orthogonality of Eigenfunctions

I recently asked this question about converting the DE $$y'' + 2y' + (\lambda + 1)y = 0$$ to Sturm-Liouville form: $$\frac{d}{dx}\left( e^{2x} y' \right) + (\lambda e^{2x} + e^{2x})y = 0$$ I am ...