PHYSICS 312
Sessional Examination, April 2000
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:
(Given again as question 2 of end of term problems set 2001)
The differential equation

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

is called the Chebychev equation.
a:
Show that the general solution to the equation can be written

$$y(x)=Af(x)+Bg(x)$$

where $f(x),g(x)$ can be expanded in a power series containing only even or odd powers.

$$f(x)=\sum_{n=0}^\infty a_nx^{2n}$$

$$g(x)=\sum_{n=0}^\infty b_nx^{2n+1}$$

b:
For what values of the constant $m$ will the series for $f(x)$ or $g(x)$ terminate so that that the solution is a polynomial with a finite number of terms? The resulting polynomials are called Chebychev polynomials $T_N(x)$ where $N$ is the highest power of $x$ in the polynomial.
c:
The Chebychev polynomials are conventionally normalized so that

$$T_N(1)=1$$

Find $T_3(x)$ and $T_4(x)$.

Problem 2:
(Given again as problem 2 of problem set 6 2001)
a:
For what values of the constants $a$ and $b$ will

$$u(r,t)=t^{b}\exp(-ar^2/t)$$

be a solution to the two dimensional heat equation

$$\nabla^2u=\frac{\partial u}{k\partial t}$$

in polar coordinates.
b:
For what values of the constants $a$ and $b$ will

$$u(r,t)=t^{b}\exp(-ar^2/t)$$

be a solution to the three dimensional heat equation

$$\nabla^2u=\frac{\partial u}{k\partial t}$$

in spherical polar coordinates.

Problem 3:
(Given again as question 3 of end of term review problem set 2001)
a:
When separating variables in spherical coordinates we encountered the differential equation

$$\frac{d}{d\theta}(\sin\theta\frac{dT}{d\theta})+\mu^2\sin\theta\;T=0$$

We transformed this equation into Legendre's equation by introducing the new variable $x=\cos\theta$. An alternative approach is to assume

$$T(\theta)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(k\theta)$$

Use the trig identity

$$\sin\theta\sin k\theta=\frac{1}{2}[\cos((k-1)\theta)-\cos((k+1)\theta))]$$

to find a recursion relation between $a_k$ and $a_{k+2}$.
b:
For what values of the constant $\mu$ will the series for the even or odd coefficients break off after a finite number of terms?

Problem 4
(Given again as question 4 of end of term review problem set 2001) Solve

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

subject to the boundary conditions

$$u(-\frac{\pi}{2},y)=u(\frac{\pi}{2},y)=u(x,-\frac{\pi}{2})=u(x,\frac{\pi}{2})=0$$

Problem 5
(Given again as problem 3 of problem set 6 2001)
a:
Find solutions to the three dimensional wave equation

$$\nabla^2u(r,t)=\frac{1}{c^2}\frac{\partial ^2u(r,t)}{\partial t^2}=0$$

with no angular dependence satisfying the condition that $u(r,t)$ is finite at $r=0$ and

$$u(R,t)=0$$

b:
Show that the three dimensional wave equation is satisfied by

$$u(r,t)=\frac{1}{r}(\phi(r-ct)+\psi(r+ct))$$

for arbitrary functions $\phi$ and $\psi$ as long as they can be differentiated twice with respect to their arguments.
c:
What are the restrictions on $\phi$ and $\psi$ in part b: for the solution to be bounded ($<\infty$) at the origin $r=0$ for all times.
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The Laplacian in plane polar coordinates can be written

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

In three dimension the Laplacian in spherical coordinates simplifies to

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

if for some reason $u$ does not depend on the angles $\theta$ and $\phi$.

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