# Sturm-Liouville Problem Cheat Sheet

• MHB
• Ackbach
In summary, the theorem states that the only solutions to the Sturm-Liouville problem are real when the condition that the function p(x)>0 fails is replaced by one of the following: classical separated boundary conditions at each endpoint, or periodic boundary conditions. The corollaries state that if one of these conditions fails, then the classical separated boundary condition at that endpoint is replaced by one that requires that y(x)<infinity at that endpoint.

#### Ackbach

Gold Member
MHB
This is a helpful document I got from one of my DE's teachers in graduate school, and I've toted it around with me. I will type it up here, as well as attach a pdf you can download.

THEOREM: Consider the Sturm-Liouville problem:
$$p(x) \, y''+p'(x) \, y'+q(x) \, y=k \, y$$
on the interval $[a,b]$ with $p(x)$ and $q(x)$ continuous, and $p(x)>0$. Assume either classical separated boundary conditions, namely, one of:
$$y(x)=0,\qquad y'(x)=0, \qquad y'(x)=c\, y(x)$$
at each endpoint $x=a$ and $x=b$; or else periodic boundary conditions:
$$y(a)=y(b), \qquad y'(a)=y'(b).$$
Then:
1. the only solutions are for $k$ real.
2. the set of all linearly independent solutions is complete for the usual
space of functions.
3. the set of linearly independent solutions is orthogonal.

COROLLARY: If the condition $p(x)>0$ fails because, at one endpoint, $p(x)=0$, then the classical separated boundary condition at that endpoint is replaced by:
$$y(x)<\infty$$
at that endpoint.

COROLLARY: If the Sturm-Liouville problem is modified by:
$$p(x) \, y''+p'(x) \, y'+q(x) \, y=k \, r(x) \, y$$
with $r(x)$ positive and continuous, and all other conditions the same, then the conclusions of the Theorem are still true, except that the orthogonality conclusion 3. is replaced by a "weighted orthogonality" with weighting function $r(x)$:
$$\int_a^b y_n(x) \, y_m(x) \, r(x) \, dx=0 \quad \text{whenever} \quad n\not=m.$$

Example 1: $y''=ky$ on $[0,\ell]$ with $y(0)=0, \; y(\ell)=0$. Then there are solutions only for $k=-\lambda_n^2=-\dfrac{n^2 \pi^2}{\ell^2}, \; n=1,2,3,\dots$ and $y_n=c_n \, \sin\left(\dfrac{n \pi x}{\ell} \right)$. This gives the Sine series.

Example 2: $y''=ky$ on $[0,\ell]$ with $y'(0)=0, \; y'(\ell)=0$. Then there are solutions for $k=-\lambda_n^2=-\dfrac{n^2 \pi^2}{\ell^2}, \; n=1,2,3,\dots$ and for $k=0$. In the second case, $y=1$ and for $n=1,2,3,\dots, \; y_n=c_n \, \cos\left(\dfrac{n \pi x}{\ell} \right)$. This gives the Cosine series.

Example 3: $y''=ky$ on $[a,b]$ with $y(a)=y(b), \; y'(a)=y'(b)$. Then there are solutions for $k=-\lambda_n^2=-\dfrac{4 n^2 \pi^2}{(b-a)^2}, \; n=1,2,3,\dots$ as well as $k=0$. Further, for each $n\ge 1$ there are two linearly independent solutions, $y_n=a_n \, \sin\left(\dfrac{2n\pi x}{b-a}\right)$ and $y_n=b_n \, \cos\left(\dfrac{2n \pi x}{b-a}\right)$. For $k=0, \; y_0=a_0$. This gives the Fourier series.

Example 4: $xy''+y'=kxy$ with $y(0)<\infty$ and $y(\ell)=0$. There are solutions for certain $k_n=-\lambda_n^2<0, \; n=1,2,3,\dots$ Call the solutions $g_n(x)$. Then the functions $g_n(x)$ are complete on the interval $[0,\ell]$ and are
orthogonal with respect to the weight $x$:
$$\int_0^{\ell}g_m(x) \, g_n(x) \, x \, dx=0 \quad \text{whenever} \quad n\not=m.$$
It turns out that $g_n(x)=J_0(\lambda_n x)$ for the zeroth Bessel function $J_0(x)$.

Attached file: https://www.physicsforums.com/attachments/4725

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Hope this will be helpful for students. Thanks for sharing!

• Ackbach