PHYSICS 312
Sessional Examination, April 2002

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: Up to 4 sheets of formulas. Calculator.

Problem 1:
Consider the differential equation

$$\frac{d^2y}{dx^2}-2x\frac{dy}{dx}-2y=0$$

a: Find a solution $y(x)$ with the initial condition

$$y(0)=1,\;\;\frac{dy}{dx}\vert _{x=0}=0$$

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? Substitute into the differential equation and verify that you have the correct solution.
c: Find a second solution with the initial condition

$$y(0)=0,\;\;\frac{dy}{dx}\vert _{x=0}=1$$

in terms of a power series

$$y(x)=\sum_{n=1}^\infty b_nx^n$$

Problem 2:
a: Find the general solution to the fourth order differential equation

$$a^4\frac{d^4u}{d\; x^4}-\omega^2 u=0$$

b: The transverse vibrations of a thick rod can be shown to satisfy the fourth order partial differential equation

$$a^4\frac{\partial ^4u}{\partial x^4}+\frac{\partial ^2u}{\partial t^2}=0$$

Find solutions on the separated variable form for the vibrations.
c: If the ends are clamped down both $u$ and $\partial u/\partial x$ vanish. Find an equation that the frequencies of vibration must satisfy for a rod of length $L$ with both ends clamped down. (you are not required to actually solve the equation). Problem 3:
When studying radioactive decay one may encounter the heat equation on the form

$$k\frac{\partial ^2u}{\partial x^2}-\frac{\partial u}{\partial t}=a\exp(-\lambda t)$$

where $a\;\lambda,\;k$ are constants, and the inhomogeneous term on the right hand side comes from heat produced by the decaying atoms. For simplicity let us assume that $k=a=1$ and that $\lambda\neq 1$
a: Attempt a particular solution

$$u(x,t)=e^{-\lambda t}v(x)$$

and find a differential equation that $v(x)$ must satisfy.
b: Solve the differential equation for $v(x)$ subject to the boundary value condition

$$v(0)=v(L)=0$$

c:
Discuss how you would solve the boundary value problem

$$\frac{\partial ^2u}{\partial x^2}-\frac{\partial u}{\partial t}=\exp(-\lambda t)$$

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

where $f(x)$ is a given function.
Problem 4:
A vibrating string of length $L$ is weakly damped by drag forces. Assume a differential equation

$$\frac{\partial ^2 u}{\partial t^2}+2f\frac{\partial u}{\partial t}-c^2\frac{\partial ^2u} {\partial x^2}=0$$

where $f>0$ and $c$ are constants.
a:
Solve the differential equation using the separation variables method assuming as boundary condition that the amplitude $u$ is zero at $x=0$ and $x=L$ and that initially

$$u(x,0)=g(x),\; \frac{d}{dt}u(x,t)\vert _{t=0}=0$$

where $g(x)$ is a known function.
b: Will the modes of high frequency die out more rapidly than the low frequency modes?
c: What is the maximum value of the damping constant for oscillations to occur at the different frequencies.
Problem 5
The general solution to the 2-dimensional Laplace equation

$$\nabla^2u(r,\theta)=\frac{\partial ^2u}{\partial r^2}+\frac{1... ...rtial r}+\frac{1}{r^2}\frac{\partial ^2 u}{\partial \theta^2}=0$$

in polar coordinates is

$$u=\alpha_0+\beta_0 \ln r +\sum_{n=1}^\infty(a_n\cos(n\theta)+b_n\sin(n\theta)) (\frac{\alpha_n}{r^n}+\beta_n r^n)$$

Find the solution to the Laplace equation in polar coordinates

$$\nabla^2u(r,\theta)=0$$

in the region $a<r<2a$ if
a:

$$u(a,\theta)=1,\;u(2a,\theta)=2$$

b:

$$u(a,\theta)=\cos\theta,\;u(2a,\theta)=\cos\theta$$

c:Find the solution to

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

in the region $a<r<2a$ if $u(a,\theta)=1,\;u(2a,\theta)=2$