PHYSICS 312
Sessional Examination, April 2001

Answer part a: of 4 out of the 5 questions and parts b: and c: of 3 questions. If more problems are attempted credit will be given for the best answers in each class. Allowed aids: 4 sheets of formulas. Calculator.

Problem 1:
(Given again as question 1 of problem set 6 2002.)
Consider the differential equation

$$2x\frac{d^2y}{dx^2}+(1-2x)\frac{dy}{dx}-y=0$$

a: Find a solution $y(x)$ with the boundary condition $y(0)=1$ as a power series

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

b: Do you recognize this power series as that of a familiar function? What is it?
c: Show that the differential equation also admits a solution

$$y=x^{1/2}\sum^\infty_{n=0}b_nx^n$$

by finding a recursion relation for the coefficients $b_n$

Problem 2:
(A modified version is given again as problem 1 of problem set 4 2002.)
Two identical metal bars are each of length a. Initially one is at temperature $0^oC$ while the other is at temperature $100^oC$. They are joined together end to end and the assembly thermally isolated. Assume the temperature satisfies the heat equation

$$k\frac{\partial ^2 T}{\partial x^2}=\frac{\partial T}{\partial t}$$

with boundary conditions

$$\frac{\partial T(x,t)}{\partial x}=0,\;\; x=-a\;\;and \;x=a$$

a:
Find the steady state solution and formulate the boundary value problem for the transient.
b:
Find the transient $u(x,t)=T(x,t)-T_S(x)$ in the form of a Fourier series.
c:
Evaluate the Fourier coefficients of the transient.

Problem 3:
(A modified version was given again as question 1 of problem set 5 2002.)
The Laplacian in spherical coordinates can be written

$$\nabla^2 u=\frac{1}{r^2}\frac{\partial }{\partial r}(r^2\frac{\partial u}{\partial r})$$

if for some reason $u$ does not depend on the angles $\theta$ and $\phi$. Assume that $u$ is independent of the angles and solve

$$\nabla^2u=-1$$

with the boundary condition
a: $u(0)=0$
b:

$$r\frac{du}{dr}\vert _{r=a}=-u$$

and $u$ bounded for $r=0$.
c:

$$u(0)=-1$$

Problem 4
(Given again as question 2 of problem set 6 2002.)
A compact way of representing the Legendre polynomials is by Rodriguez formula

$$P_n(x)=\frac{1}{n!2^n}\frac{d^n}{dx^n}[(x^2-1)^n]$$

To make this result plausible
a: compute $P_1(x),P_2(x)$ and $P_3(x)$ using the formula
b: verify that $P_1(x),P_2(x)$ and $P_3(x)$ computed above satisfies

$$P_n(1)=1$$

c: Show that $P_1(x),P_2(x)$ and $P_3(x)$ computed above satisfies Legendre's differential equation

$$(1-x^2)\frac{d^2P_n(x)}{dx^2}-2x\frac{dP_n(x)}{dx}+n(n+1) P_n(x)=0$$ (1)

for the appropriate $n$.

Problem 5
(Given again as question 3 of problem set 6 2002.)
When considering heat flow around a flat circular disk with convection one encounters the modified Bessel equation of order zero.

$$\frac{d}{dr}(r\frac{d\rho}{dr})-(\frac{n^2}{r}+r)\rho=0$$

The solutions to this equation, that are bounded at the origin, are called modified Bessel functions of order $n$ and written $I_n(r)$. In this problem we restrict our attention to the case $n=0$
a: Assume that $I_0=1$ and make a power series expansion

$$I_0(r)=\sum_{m=0}^\infty a_mr^m$$

Find a recursion formula relating $a_m$ to $a_{m+2}$
b: Find $a_m$ for $m=2,4$
c: Show that $a_1=0$ if the solution is bounded for $r=0$.

END OF EXAMINATION


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