PHYSICS 312, Midterm Examination, February 26 1999
Answer all three problems

Problem 1: In a heat conduction problem the steady state temperature T_S satisfies the differential equation

$$\frac{d^2T_S}{dx^2}=h^2(T_S-T_0-bx)$$

where T₀ and b are constants and the boundary conditions are

T_S(0)=T_S(a)=0

find T_S.

Problem 2:

a:

For which values of $\lambda$ will

$$\frac{d^2\phi}{dx^2}+2\frac{d\phi}{dx}+\lambda^2\phi=0$$

have solutions satisfying

$$\phi(0)=\phi(\pi)=0$$

b:

Sketch the eigenfunctions corresponding to the three lowest eigenvalues.

Problem 3:

a:

Find a formula for the solution to the problem

$$\frac{\partial^2u}{\partial x^2}=\frac{1}{k}\frac{\partial u}{\partial t},\; 0<x<\infty,\;0<t$$

$$u(x,0)=f(x),\;\;u(0,t)=0$$

b:

Find an explicit solution in the special case

$$f(x)=T_0\sin(x)$$

where T₀ is a constant.

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