PHYSICS 312
Sessional Examination, April 1999
Answer 4 out of the 5 problems. All problems have equal value. If all 5 problems are attempted credit will be given for the best 4 answers.

Problem 1:
Find the steady state solution of the problem

$$\frac{\partial}{\partial x}\left((b+fx)\frac{\partial u}{\partial x}\right)=\frac{1}{k} \frac{\partial u}{\partial t}$$

$$0<x<a,\;0<t$$

$$u(0,t)=T_1;\;u(a,t)=T_1,0<t$$

b,f,k are constants.
(This problem was given again as question 1 of midterm review problem set 2001.)).

Problem 2:
Find the six lowest frequencies of a vibrating membrame satisfying

$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=\frac{1}{c^2}\frac {\partial^2 u}{\partial t^2}$$

The membrane is rectangular in shape with sides a and 2a, and u=0 on the boundary.

Problem 3:
The Laplacian in plane polar coordinates can be written

$$\nabla^2u=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\part... ...partial r}) +\frac{1}{r^2}\frac{\partial^2u}{\partial\theta^2}$$

a:
Find a particular solution to

$$\nabla^2u=r$$

b:
Solve the boundary value problem

$$\nabla^2u=r$$

$$u(1,\theta)=\sin\theta$$

in the region $0\leq r\leq a.$


Problem 4:
The differential equation

$$y^{\prime\prime}(x)-2xy^\prime(x)+2\alpha y(x)=0$$

is called Hermite's equation ( $\alpha=constant$).

a:
Find a recursion formula for the coefficients a_n of the power series solution

$$y(x)=\sum_{n=0}^\infty a_nx^n$$

subject to an initial condition

$$y(0)=a_0,\;\;y^\prime(x)\vert _{x=0}=a_1$$

b:
For what initial conditions and values of $\alpha$ will there be solutions to Hermite's equation that are polynomials of order n

$$y=a_0+a_1x+a_2 x^2 \cdots a_nx^n$$

Show that the polynomial will contain either only odd or only even power of n.

c:
Find polynomials of order n=3 and n=4 that are solutions to Hermite's equation.

Problem 5:
Solve the boundary value problem

$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$

inside a square of side $2\pi$:

$$u=(x,-\pi)=u(x,\pi)=\sin x;\;u(\pi,y)=u(\pi,y)=\sin y$$

(This problem was given again as question 2 of midterm review problem set 2001.)).

END OF EXAMINATION

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