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 $$\mathscr{L}y=\lambda y$$

status measured difficulty not learned 37% [default] 0

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>