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\title{m-3150-001 ~~~ Midterm exam 2}
\author{Andrej Cherkaev}
\date{}
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Your name, ...........................................................

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Solve any three problems: 
\begin{enumerate}
\item Find the displacement $u(x, t)$ of a string of a length 5, if its initial displacement $u(x,0)$ is 
$$
u(x,0)= \sin \left(\frac{\pi x}{5} \right) + \frac{1}{3} \sin \left(\frac{3\pi x}{5} \right),   
$$
 its initial speed is zero. 
and the spring constant $c$ is $c=2$.
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\item Initially, a rod has zero temperature 
$
u(x,0)=0  
$.
Its left end ($x=0$) is kept at zero temperature $u(0, t)=0$ and its right end ($x=2$) -- at the temperature 100,    $u(2, t)=100$, and the diffusivity constant $c$ is equal to one, $c=1$. 
Find the temperature inside the rod at $t=2$. 
(Hint: Account for nonzero boundary conditions) 
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\item
The left end of a rod is thermo-insulated, and the right end is kept at zero temperature. 
$$
\left.\frac{\partial u(x,t)}{\partial x} \right|_{x=0} =0,   \quad u(3, t)=0,  \quad \forall t \in [0, \infty).
$$
Find a general solution to the heat problem, 
assuming that the initial temperature is given, $u(x, 0)=f(x)$  and the diffusivity constant $c$ is equal to one, $c=1$. 
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\item

Separate variables in the the two-dimensional heat equation {\it (proceed similarly to the two-dimensional wave equation analysis)}. Show the scheme of variables separation,  resulting differential differential equations, general soultion,  do {\it not} compute coefficients. 
$$
\frac{\partial u(x,y,t)}{\partial t}= c^2\left(  \frac{\partial^2 u(x,y,t)}{\partial x^2}+ \frac{\partial^2 u(x,y,t)}{\partial y^2}\right)
$$
if
$0 \leq x\leq a, ~~ 0 \leq y \leq b,$  $u(x, y, 0)= f(x, y)$,

 $u(0, y, t)=u(a, y, t)=0$, 

$u(x,0, t)=u(x, b, t)=0$ .

 
\end{enumerate} 

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