Part IA, 2004, Paper 2

Differential Equations

• 2.I.1B

  Differential Equations | Part IA, 2004

  By writing $y(x)=m x$ where $m$ is a constant, solve the differential equation

  $$\frac{d y}{d x}=\frac{x-2 y}{2 x+y}$$

  and find the possible values of $m$.

  Describe the isoclines of this differential equation and sketch the flow vectors. Use these to sketch at least two characteristically different solution curves.

  Now, by making the substitution $y(x)=x u(x)$ or otherwise, find the solution of the differential equation which satisfies $y(0)=1$.

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• 2.I.2B

  Differential Equations | Part IA, 2004

  Find two linearly independent solutions of the differential equation

  $$\frac{d^{2} y}{d x^{2}}+2 p \frac{d y}{d x}+p^{2} y=0$$

  Find also the solution of

  $$\frac{d^{2} y}{d x^{2}}+2 p \frac{d y}{d x}+p^{2} y=e^{-p x}$$

  that satisfies

  $$y=0, \frac{d y}{d x}=0 \quad \text { at } x=0$$

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• 2.II.5B

  Differential Equations | Part IA, 2004

  Construct a series solution $y=y_{1}(x)$ valid in the neighbourhood of $x=0$, for the differential equation

  $$\frac{d^{2} y}{d x^{2}}+4 x^{3} \frac{d y}{d x}+x^{2} y=0$$

  satisfying

  $$y_{1}=1, \frac{d y_{1}}{d x}=0 \quad \text { at } x=0 .$$

  Find also a second solution $y=y_{2}(x)$ which satisfies

  $$y_{2}=0, \frac{d y_{2}}{d x}=1 \quad \text { at } x=0 .$$

  Obtain an expression for the Wronskian of these two solutions and show that

  $$y_{2}(x)=y_{1}(x) \int_{0}^{x} \frac{e^{-\xi^{4}}}{y_{1}^{2}(\xi)} d \xi$$

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• 2.II.6B

  Differential Equations | Part IA, 2004

  Two solutions of the recurrence relation

  $$x_{n+2}+b(n) x_{n+1}+c(n) x_{n}=0$$

  are given as $p_{n}$ and $q_{n}$, and their Wronskian is defined to be

  $$W_{n}=p_{n} q_{n+1}-p_{n+1} q_{n}$$

  Show that

  $$W_{n+1}=W_{1} \prod_{m=1}^{n} c(m)$$

  Suppose that $b(n)=\alpha$, where $\alpha$ is a real constant lying in the range $[-2,2]$, and that $c(n)=1$. Show that two solutions are $x_{n}=e^{i n \theta}$ and $x_{n}=e^{-i n \theta}$, where $\cos \theta=-\alpha / 2$. Evaluate the Wronskian of these two solutions and verify $(*)$.

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• 2.II.7B

  Differential Equations | Part IA, 2004

  Show how a second-order differential equation $\ddot{x}=f(x, \dot{x})$ may be transformed into a pair of coupled first-order equations. Explain what is meant by a critical point on the phase diagram for a pair of first-order equations. Hence find the critical points of the following equations. Describe their stability type, sketching their behaviour near the critical points on a phase diagram.

  $$\begin{aligned} &\text { (i) } \ddot{x}+\cos x=0 \\ &\text { (ii) } \ddot{x}+x\left(x^{2}+\lambda x+1\right)=0, \quad \text { for } \lambda=1,5 / 2 . \end{aligned}$$

  Sketch the phase portraits of these equations marking clearly the direction of flow.

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• 2.II.8B

  Differential Equations | Part IA, 2004

  Construct the general solution of the system of equations

  $$\begin{aligned} &\dot{x}+4 x+3 y=0 \\ &\dot{y}+4 y-3 x=0 \end{aligned}$$

  in the form

  $$\left(\begin{array}{l} x(t) \\ y(t) \end{array}\right)=\mathbf{x}=\sum_{j=1}^{2} a_{j} \mathbf{x}^{(j)} e^{\lambda_{j} t}$$

  and find the eigenvectors $\mathbf{x}^{(j)}$ and eigenvalues $\lambda_{j}$.

  Explain what is meant by resonance in a forced system of linear differential equations.

  Consider the forced system

  $$\begin{aligned} &\dot{x}+4 x+3 y=\sum_{j=1}^{2} p_{j} e^{\lambda_{j} t} \\ &\dot{y}+4 y-3 x=\sum_{j=1}^{2} q_{j} e^{\lambda_{j} t} \end{aligned}$$

  Find conditions on $p_{j}$ and $q_{j}(j=1,2)$ such that there is no resonant response to the forcing.

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Probability

• 2.I.3F

  Probability | Part IA, 2004

  Define the covariance, $\operatorname{cov}(X, Y)$, of two random variables $X$ and $Y$.

  Prove, or give a counterexample to, each of the following statements.

  (a) For any random variables $X, Y, Z$

  $$\operatorname{cov}(X+Y, Z)=\operatorname{cov}(X, Z)+\operatorname{cov}(Y, Z)$$

  (b) If $X$ and $Y$ are identically distributed, not necessarily independent, random variables then

  $$\operatorname{cov}(X+Y, X-Y)=0$$

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• 2.I.4F

  Probability | Part IA, 2004

  The random variable $X$ has probability density function

  $$f(x)= \begin{cases}c x(1-x) & \text { if } 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise }\end{cases}$$

  Determine $c$, and the mean and variance of $X$.

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• 2.II.10F

  Probability | Part IA, 2004

  Define the conditional probability $P(A \mid B)$ of the event $A$ given the event $B$.

  A bag contains four coins, each of which when tossed is equally likely to land on either of its two faces. One of the coins shows a head on each of its two sides, while each of the other three coins shows a head on only one side. A coin is chosen at random, and tossed three times in succession. If heads turn up each time, what is the probability that if the coin is tossed once more it will turn up heads again? Describe the sample space you use and explain carefully your calculations.

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• 2.II.11F

  Probability | Part IA, 2004

  The random variables $X_{1}$ and $X_{2}$ are independent, and each has an exponential distribution with parameter $\lambda$. Find the joint density function of

  $$Y_{1}=X_{1}+X_{2}, \quad Y_{2}=X_{1} / X_{2}$$

  and show that $Y_{1}$ and $Y_{2}$ are independent. What is the density of $Y_{2}$ ?

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• 2.II.12F

  Probability | Part IA, 2004

  Let $A_{1}, A_{2}, \ldots, A_{r}$ be events such that $A_{i} \cap A_{j}=\emptyset$ for $i \neq j$. Show that the number $N$ of events that occur satisfies

  $$P(N=0)=1-\sum_{i=1}^{r} P\left(A_{i}\right)$$

  Planet Zog is a sphere with centre $O$. A number $N$ of spaceships land at random on its surface, their positions being independent, each uniformly distributed over the surface. A spaceship at $A$ is in direct radio contact with another point $B$ on the surface if $\angle A O B<\frac{\pi}{2}$. Calculate the probability that every point on the surface of the planet is in direct radio contact with at least one of the $N$ spaceships.

  [Hint: The intersection of the surface of a sphere with a plane through the centre of the sphere is called a great circle. You may find it helpful to use the fact that $N$ random great circles partition the surface of a sphere into $N(N-1)+2$ disjoint regions with probability one.]

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• 2.II.9F

  Probability | Part IA, 2004

  Let $X$ be a positive-integer valued random variable. Define its probability generating function $p_{X}$. Show that if $X$ and $Y$ are independent positive-integer valued random variables, then $p_{X+Y}=p_{X} p_{Y}$.

  A non-standard pair of dice is a pair of six-sided unbiased dice whose faces are numbered with strictly positive integers in a non-standard way (for example, $(2,2,2,3,5,7$ ) and $(1,1,5,6,7,8))$. Show that there exists a non-standard pair of dice $A$ and $B$ such that when thrown

  $P\{$ total shown by $A$ and $B$ is $n\}=P\{$ total shown by pair of ordinary dice is $n\}$

  for all $2 \leqslant n \leqslant 12$.

  [Hint: $\left.\left(x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}\right)=x(1+x)\left(1+x^{2}+x^{4}\right)=x\left(1+x+x^{2}\right)\left(1+x^{3}\right) .\right]$

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