## 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 $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
1. Show that the Green's function for this problem is $G(x,y)=\left\{ \begin{array}{rl} -(2y-1)x, & 0 \le x < y \le 1\\ -(2x-1)y, & 0 \le y< x \le 1. \end{array} \right.$
2. Verify that $0$ is not an eigenvalue for $Kf(x) := \int_0^1G(x,y)f(y)dy$.
3. 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.
4. 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.