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\centerline{{\sc Problem set 6}}
\centerline{Math 207B, Winter 2016}
\centerline{Due: Fri, Mar. 4}


\bigskip\noindent
{\bf 1.} Suppose that $u_1, u_2 : \Rl \to \Rl$ are two solutions of the homogeneous Sturm-Liouville equation
\[
-(pu')' + q u = 0
\]
where $p, q: \Rl \to \Rl$ are smooth functions and $p>0$.
If $W = u_1 u_2' - u_2 u_1'$
is the Wronskian of $u_1$, $u_2$, show that
$p W = \text{constant}$.


\bigskip\noindent
{\bf 2.} Compute the Green's function for the BVP
\begin{align*}
&-u'' + u = f(x)\qquad 0<x<1
\\
&u(0)=0,\qquad u(1)=0.
\end{align*}
Write down the integral representation of the solution $u$ in terms of $f$.

\bigskip\noindent
{\bf 3.} Compute the Green's function for the BVP
\begin{align*}
&-u'' = f(x)\qquad 0<x<1
\\
&u(0)+u(1)=0,\qquad u'(0)+u'(1)=0.
\end{align*}
Write down the integral representation of the solution $u$ in terms of $f$.


\bigskip\noindent
{\bf 4.} Compute the generalized Green's function $G(x,\xi)$ for the BVP
\begin{align*}
&-u'' = \pi^2 u + f(x)\qquad 0<x<1
\\
&u(0) = 0,\qquad u(1)=0.
\end{align*}
State the equations that are satisfied by the function
\[
u(x) = \int_0^1 G(x,\xi) f(\xi)\, d\xi.
\]


\newpage\noindent
{\bf 5.} Consider the Sturm-Liouville equation
\[
-(pu')' + q u = \lambda r u,\qquad a<x<b
\]
where $p, q, r: [a,b] \to \Rl$ are smooth functions and $p(x), r(x) > 0$ for $a\le x\le b$.
Show that the change of variables
\[
t = \int_a^x \sqrt{\frac{r(s)}{p(s)}}\, ds,\qquad v(t) = \left[r(x) p(x)\right]^{1/4} u(x)
\]
transforms this equation into a  Sturm-Liouville equation with $p=r=1$ of the form
\[
-v'' + Q v = \lambda v,\qquad 0<t<c.
\]
What are $c$ and $Q:[0,c]\to \Rl$?



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