Edited, memorised or added to reading queue

on 22-Nov-2023 (Wed)

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Annotation 7598437960972

Sturm–Liouville Problem

In mathematics and its applications, a Sturm-Liouville problem is a second-order linear ordinary differential equation of the form:\(\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}\right]+q(x) y=-\lambda w(x) y\) for given functions \(p(x), q(x)\) and \(w(x)\), together with some boundary conditions at extreme values of \(x\).

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Flashcard 7598439795980

Question

In mathematics and its applications, a Sturm-Liouville problem is a second-order linear ordinary differential equation of the form:[...]

Answer
\(\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}\right]+q(x) y=-\lambda w(x) y\) for given functions \(p(x), q(x)\) and \(w(x)\), together with some boundary conditions at extreme values of \(x\).

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Sturm–Liouville Problem
In mathematics and its applications, a Sturm-Liouville problem is a second-order linear ordinary differential equation of the form:\(\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}\right]+q(x) y=-\lambda w(x) y\) for given functions \(p(x), q(x)\) and \(w(x)\), together with some boundary conditions at extreme values of \(x\).






Annotation 7598441368844

Defintion of the symmetric/self-adjoint operator of S-L system
For the S-L system \(\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}\right]+q(x) y=-\lambda w(x) y\), we can define a symmetric/self-adjoint operator of S-L system: \(\mathscr{L}=\frac{-1}{r}\left[\frac{\mathrm{d}}{\mathrm{d} x}\left(p(x) \frac{\mathrm{d}}{\mathrm{~d} x}\right)+q(x)\right]\) and transform the original equation to \(\mathscr{L}y=\lambda y\)
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Flashcard 7598443203852

Question
For the S-L system \(\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}\right]+q(x) y=-\lambda w(x) y\), we can define a symmetric/self-adjoint operator of S-L system: [...]
Answer
\(\mathscr{L}=\frac{-1}{r}\left[\frac{\mathrm{d}}{\mathrm{d} x}\left(p(x) \frac{\mathrm{d}}{\mathrm{~d} x}\right)+q(x)\right]\) and transform the original equation to \(\mathscr{L}y=\lambda y\)

statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Defintion of the symmetric/self-adjoint operator of S-L system
For the S-L system \(\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}\right]+q(x) y=-\lambda w(x) y\), we can define a symmetric/self-adjoint operator of S-L system: <span>\(\mathscr{L}=\frac{-1}{r}\left[\frac{\mathrm{d}}{\mathrm{d} x}\left(p(x) \frac{\mathrm{d}}{\mathrm{~d} x}\right)+q(x)\right]\) and transform the original equation to \(\mathscr{L}y=\lambda y\) <span>