Jan. 18, 1996 \hfill Math 312, Quiz 1 \hfill Name:

\q{5}Define the operator $\ds L[u]=\frac{\p u}{\p t}-\frac{\p^2u}{\p
x^2}$.

\item Show that $L$ is a linear operator.


\item Show that $\ds u_n(x,t)=\sin(nx)e^{-n^2t}$ satisfies
$L[u_n]=0$. Thus, from part a., we know that any function of the form 
$$
u(x,t)=\sum_{n=1}^Nb_n\sin(nx)e^{-n^2t}
$$
is a solution to the differential equation: $L[u]=0$.

\q{5}Are there any nontrivial solutions of the form $u(x,y)=X(x)Y(y)$
to the problem: 

\frac{\p^3 u}{\p x^2\p y}&=&0, \mbox{   for }0 < x < 1,\mbox{ and }y>0,
X(0)&=&0,\ y>0,
X(1)&=&0,\ y>0.


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