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.

# Questions tagged [sturm-liouville]

395 questions

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### Prove that all the eigenvalues of the following Sturm–Liouville problem are positive: $u'' + (\lambda -x^2)u = 0$ $\hspace{0.5cm}$ $0< x < \infty$

Prove that all the eigenvalues of the following Sturm–Liouville problem are
positive
$u'' + (\lambda -x^2)u = 0$ $\hspace{0.5cm}$ $0< x < \infty$
$u'(0) = \lim_{x \to \infty}{u(x)} =0$
I'm trying to solve this Sturm-Liouville problem. But I haven't…

Jack

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### Can a D.E. still be defined even if terms are unbounded?

$$(1 - x^2)y'' - 2xy' + l(l + 1)y = 0, -1 \leq x \leq 1$$
I want to go ahead and use the Sturm-Liouville theorem to prove that this equation's eigenstates are orthogonal, but it's not a given that $y'$ is bounded at $a$ or $b$, only that the D.E. is…

Λάιος Ζαφειρίου

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### Sturm-Lioville problem: Term by term differentiation of eigenfunction expansion

Say I want to solve an Sturm-Lioville problem with non-homogeneous term but homogeneous boundary conditions:
$y''+(1/x)y'+(1/x^2)y=x$
$y'(0)=0$
$y(1)=0$
One method of solution involves finding the eigenfunctions associated with the homogeneous…

user3199900

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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:
$$ y (0) + y'(0) = 0 $$
$$ y(1) + y'(1) = 0 $$
So my attempt at this goes something like…

Safder

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### Is 2nd-order ODE with quadratic coefficients solvable?

Consider an ODE eigensystem
$$\begin{bmatrix}
0 & d_1-\mathrm id_2 \\
d_1+\mathrm id_2 & 0
\end{bmatrix}
\begin{bmatrix} a(y) \\ b(y) \end{bmatrix} = \lambda \begin{bmatrix} a(y) \\ b(y) \end{bmatrix},
$$
where $$d_1=-\mathrm…

xiaohuamao

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### Negative eigenvalues of Sturm-Liouville problem

I am trying to solve the following Sturm-Liouville problem on the circle $S^1$, i.e. the closed interval $[0,1]$ with periodic boundary conditions $(y(0)=y(1))$:…

smnas

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### Higher order Sturm-Liouville form

The differential equation book I was reading briefly mentions about the generalization S-L form and it says if we consider the BVP
$L(y)=\lambda r(x)y$
where $L(y)=\displaystyle P_n(x)\frac{d^ny}{dx^n}+\dots +P_1(x)\frac{dy}{dx}+P_0(x)y$, then the…

J.Shim

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### A question about Sturm Liouville of transforming it into normal form

I have a question about Sturm Liouville problem.
Given a SL equation, say $\frac{d^2 y}{dx^2}+p(x)\frac{d y}{dx}+(q(x)+\lambda r(x))y=0$, how to transform it to its normal form $\frac{d^2 \eta}{d\xi^2}+(\phi(\xi)+\lambda)\eta=0$? How one can think…

will_cheuk

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### How guarantee that the Sturm-Liouville spectrum is discrete?

Suppose that we want to series expand a function $f \in L^2_w(0,\infty)$ in terms of eigenfunctions of a given Sturm-Liouville system having weight function $w$ for the half line, and that we have already found that the left end ($0$) is regular,…

Martin

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### Liouville's equation and Cauchy problem $\dot{x} = f(x(t),t)$ in optimal control

In a guest lecture about optimal control problem, the speaker introduce the following:
The Liouville's equation encodes a superposition al all classical solutions soving Cauchy problem:
Liouville's equation: $$\frac{\partial \mu}{\partial t}+…

sleeve chen

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### Questions about the proof of the Sturm oscillation theorem

I'm trying to understand the proof of the Sturm oscillation theorem and I hit the roadblock.
Theorem: Let $E_0

Konstantin

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### Is this Set of Bessel Functions a Basis for All $C^{1}[0,a]$ Functions?

Consider the following set of Bessel functions
$$\{J_1(\alpha_ir)\}, \qquad J_0(\alpha_ia)=0 \tag{1}$$
I want to show that this set of functions form a basis for the space of $C^{1}[0,a]$ functions. So I should prove that
They are linearly…

Hosein Rahnama

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### Sturm-Liouville-Problem with confusing result

I have the following ODE defined on $D_x = [-1,1]$:
$$y''(x)=-k^2y(x).$$
Prom the physical problem, I know that the solution is non-zero and that $y(x)$ is a even function [$y(-x)=y(x)$] vanishing at $x=1$.
The general solution to the previous ODE…

MrYouMath

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### meaning of in-homogenous term in the wave function

Suppose I have a string of mass density $\rho$ clumped between two points so that the tension is $T$.
The resonant standing waves of this vibrating string are those in which the restoring force on the elements of the string are proportional to their…

E Be

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### Sturm Liouville eigenvalue problem

I have the following Sturm-Liouville eigenvalue problem,
$$\frac{d}{dr}\left(r \frac{d u}{dr} \right) + k^2 r u = 0, \, \, u(R) = 0, \, u(0) < \infty$$
where $k^2$ the eigenvalues and $u(r)$ the associated eigenfunctions. I was able to prove that…

stokes

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