## Math 641-600 Suggested Problems

**Extra Problems** - These are *not* to be handed
in.

- Section 4.1: 1(b), 4, 6
- Section 4.2: 1, 4, 8
- Section 4.3: 3
- Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
- Show that the Green's function
for this problem is
\[
G(x,y)=\left\{
\begin{array}{rl}
-(2y-1)x, & 0 \le x < y \le 1\\
-(2x-1)y, & 0 \le y< x \le 1.
\end{array} \right.
\]
- Verify that $0$ is not an eigenvalue for $Kf(x) :=
\int_0^1G(x,y)f(y)dy$.
- Verify that $u=x^2$ satisfies the boundary conditions required
for it to be in the domain of $L$ and that $\langle Lu,u \rangle <
0$. Also verify if we let $u=x(x-1)^2$, then $u$ is in the domain
for $L$ and that $\langle Lu,u \rangle > 0$. Thus $L$ does not
satisfy the conditions of Keener's Theorem 4.7.
- Even though the Theorem 4.7 is not satisfied, the orthonormal set
of eigenfunctions for $L$ form a complete set in $L^2[0,1]$,
Explain.

Updated 12/9/14.