Part IA, 2003, Paper 2

# Part IA, 2003, Paper 2

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2.I.1D

commentConsider the equation

$\frac{d y}{d x}=1-y^{2} .$

Using small line segments, sketch the flow directions in $x \geqslant 0,-2 \leqslant y \leqslant 2$ implied by the right-hand side of $(*)$. Find the general solution (i) in $|y|<1$, (ii) in $|y|>1$.

Sketch a solution curve in each of the three regions $y>1,|y|<1$, and $y<-1$.

2.I.2D

commentConsider the differential equation

$\frac{d x}{d t}+K x=0$

where $K$ is a positive constant. By using the approximate finite-difference formula

$\frac{d x_{n}}{d t}=\frac{x_{n+1}-x_{n-1}}{2 \delta t}$

where $\delta t$ is a positive constant, and where $x_{n}$ denotes the function $x(t)$ evaluated at $t=n \delta t$ for integer $n$, convert the differential equation to a difference equation for $x_{n}$.

Solve both the differential equation and the difference equation for general initial conditions. Identify those solutions of the difference equation that agree with solutions of the differential equation over a finite interval $0 \leqslant t \leqslant T$ in the limit $\delta t \rightarrow 0$, and demonstrate the agreement. Demonstrate that the remaining solutions of the difference equation cannot agree with the solution of the differential equation in the same limit.

[You may use the fact that, for bounded $|u|, \quad \lim _{\epsilon \rightarrow 0}(1+\epsilon u)^{1 / \epsilon}=e^{u}$.]

2.II.5D

comment(a) Show that if $\mu(x, y)$ is an integrating factor for an equation of the form

$f(x, y) d y+g(x, y) d x=0$

then $\partial(\mu f) / \partial x=\partial(\mu g) / \partial y$.

Consider the equation

$\cot x d y-\tan y d x=0$

in the domain $0 \leqslant x \leqslant \frac{1}{2} \pi, \quad 0 \leqslant y \leqslant \frac{1}{2} \pi$. Using small line segments, sketch the flow directions in that domain. Show that $\sin x \cos y$ is an integrating factor for the equation. Find the general solution of the equation, and sketch the family of solutions that occupies the larger domain $-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi,-\frac{1}{2} \pi \leqslant y \leqslant \frac{1}{2} \pi$.

(b) The following example illustrates that the concept of integrating factor extends to higher-order equations. Multiply the equation

$\left[y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}\right] \cos ^{2} x=1$

by $\sec ^{2} x$, and show that the result takes the form $\frac{d}{d x} h(x, y)=0$, for some function $h(x, y)$ to be determined. Find a particular solution $y=y(x)$ such that $y(0)=0$ with $d y / d x$ finite at $x=0$, and sketch its graph in $0 \leqslant x<\frac{1}{2} \pi$.

2.II.7D

commentConsider the linear system

$\dot{\mathbf{x}}(t)-A \mathbf{x}(t)=\mathbf{z}(t)$

where the $n$-vector $\mathbf{z}(t)$ and the $n \times n$ matrix $A$ are given; $A$ has constant real entries, and has $n$ distinct eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ and $n$ linearly independent eigenvectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}$. Find the complementary function. Given a particular integral $\mathbf{x}_{\mathbf{p}}(t)$, write down the general solution. In the case $n=2$ show that the complementary function is purely oscillatory, with no growth or decay, if and only if

$\operatorname{trace} A=0 \quad \text { and } \quad \operatorname{det} A>0 .$

Consider the same case $n=2$ with trace $A=0$ and $\operatorname{det} A>0$ and with

$\mathbf{z}(t)=\mathbf{a}_{1} \exp \left(i \omega_{1} t\right)+\mathbf{a}_{2} \exp \left(i \omega_{2} t\right)$

where $\omega_{1}, \omega_{2}$ are given real constants. Find a particular integral when

(i) $i \omega_{1} \neq \lambda_{1}$ and $i \omega_{2} \neq \lambda_{2}$;

(ii) $i \omega_{1} \neq \lambda_{1}$ but $i \omega_{2}=\lambda_{2}$.

In the case

$A=\left(\begin{array}{cc} 1 & 2 \\ -5 & -1 \end{array}\right)$

with $\mathbf{z}(t)=\left(\begin{array}{c}2 \\ 3 i-1\end{array}\right) \exp (3 i t)$, find the solution subject to the initial condition $\mathbf{x}=\left(\begin{array}{l}1 \\ 0\end{array}\right)$ at $t=0$.

2.II.8D

commentFor all solutions of

$\begin{aligned} &\dot{x}=\frac{1}{2} \alpha x+y-2 y^{3} \\ &\dot{y}=-x \end{aligned}$

show that $d K / d t=\alpha x^{2}$ where

$K=K(x, y)=x^{2}+y^{2}-y^{4}$

In the case $\alpha=0$, analyse the properties of the critical points and sketch the phase portrait, including the special contours for which $K(x, y)=\frac{1}{4}$. Comment on the asymptotic behaviour, as $t \rightarrow \infty$, of solution trajectories that pass near each critical point, indicating whether or not any such solution trajectories approach from, or recede to, infinity.

Briefly discuss how the picture changes when $\alpha$ is made small and positive, using your result for $d K / d t$ to describe, in qualitative terms, how solution trajectories cross $K$-contours.

2.I.3F

comment(a) Define the probability generating function of a random variable. Calculate the probability generating function of a binomial random variable with parameters $n$ and $p$, and use it to find the mean and variance of the random variable.

(b) $X$ is a binomial random variable with parameters $n$ and $p, Y$ is a binomial random variable with parameters $m$ and $p$, and $X$ and $Y$ are independent. Find the distribution of $X+Y$; that is, determine $P\{X+Y=k\}$ for all possible values of $k$.

2.I.4F

commentThe random variable $X$ is uniformly distributed on the interval $[0,1]$. Find the distribution function and the probability density function of $Y$, where

$Y=\frac{3 X}{1-X}$

2.II.10F

commentThe random variables $X$ and $Y$ each take values in $\{0,1\}$, and their joint distribution $p(x, y)=P\{X=x, Y=y\}$ is given by

$p(0,0)=a, \quad p(0,1)=b, \quad p(1,0)=c, \quad p(1,1)=d .$

Find necessary and sufficient conditions for $X$ and $Y$ to be (i) uncorrelated; (ii) independent.

Are the conditions established in (i) and (ii) equivalent?

2.II.11F

commentA laboratory keeps a population of aphids. The probability of an aphid passing a day uneventfully is $q<1$. Given that a day is not uneventful, there is probability $r$ that the aphid will have one offspring, probability $s$ that it will have two offspring and probability $t$ that it will die, where $r+s+t=1$. Offspring are ready to reproduce the next day. The fates of different aphids are independent, as are the events of different days. The laboratory starts out with one aphid.

Let $X_{1}$ be the number of aphids at the end of the first day. What is the expected value of $X_{1}$ ? Determine an expression for the probability generating function of $X_{1}$.

Show that the probability of extinction does not depend on $q$, and that if $2 r+3 s \leqslant 1$ then the aphids will certainly die out. Find the probability of extinction if $r=1 / 5, s=2 / 5$ and $t=2 / 5$.

[Standard results on branching processes may be used without proof, provided that they are clearly stated.]

2.II.12F

commentPlanet Zog is a ball with centre $O$. Three spaceships $A, B$ and $C$ land at random on its surface, their positions being independent and each uniformly distributed on its surface. Calculate the probability density function of the angle $\angle A O B$ formed by the lines $O A$ and $O B$.

Spaceships $A$ and $B$ can communicate directly by radio if $\angle A O B<\pi / 2$, and similarly for spaceships $B$ and $C$ and spaceships $A$ and $C$. Given angle $\angle A O B=\gamma<\pi / 2$, calculate the probability that $C$ can communicate directly with either $A$ or $B$. Given angle $\angle A O B=\gamma>\pi / 2$, calculate the probability that $C$ can communicate directly with both $A$ and $B$. Hence, or otherwise, show that the probability that all three spaceships can keep in in touch (with, for example, $A$ communicating with $B$ via $C$ if necessary) is $(\pi+2) /(4 \pi)$.

2.II.9F

commentState the inclusion-exclusion formula for the probability that at least one of the events $A_{1}, A_{2}, \ldots, A_{n}$ occurs.

After a party the $n$ guests take coats randomly from a pile of their $n$ coats. Calculate the probability that no-one goes home with the correct coat.

Let $p(m, n)$ be the probability that exactly $m$ guests go home with the correct coats. By relating $p(m, n)$ to $p(0, n-m)$, or otherwise, determine $p(m, n)$ and deduce that

$\lim _{n \rightarrow \infty} p(m, n)=\frac{1}{e m !} .$