Part IA, 2007, Paper 2

Differential Equations

• Paper 2, Section I, B

  Differential Equations | Part IA, 2007

  Investigate the stability of:

  (i) the equilibrium points of the equation

  $$\frac{d y}{d t}=\left(y^{2}-4\right) \tan ^{-1}(y)$$

  (ii) the constant solutions $\left(u_{n+1}=u_{n}\right)$ of the discrete equation

  $$u_{n+1}=\frac{1}{2} u_{n}^{2}\left(1+u_{n}\right)$$

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• Paper 2, Section I, B

  Differential Equations | Part IA, 2007

  Find the solution $y(x)$ of the equation

  $$y^{\prime \prime}-6 y^{\prime}+9 y=\cos (2 x) \mathrm{e}^{3 x}$$

  that satisfies $y(0)=0$ and $y^{\prime}(0)=1$.

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• Paper 2, Section II, B

  Differential Equations | Part IA, 2007

  (i) Find the general solution of the difference equation

  $$u_{k+1}+5 u_{k}+6 u_{k-1}=12 k+1$$

  (ii) Find the solution of the equation

  $$y_{k+1}+5 y_{k}+6 y_{k-1}=2^{k}$$

  that satisfies $y_{0}=y_{1}=1$. Hence show that, for any positive integer $n$, the quantity $2^{n}-26(-3)^{n}$ is divisible by $10 .$

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• Paper 2, Section II, B

  Differential Equations | Part IA, 2007

  (i) Find, in the form of an integral, the solution of the equation

  $$\alpha \frac{d y}{d t}+y=f(t)$$

  that satisfies $y \rightarrow 0$ as $t \rightarrow-\infty$. Here $f(t)$ is a general function and $\alpha$ is a positive constant.

  Hence find the solution in each of the cases:

  (a) $f(t)=\delta(t)$;

  (b) $f(t)=H(t)$, where $H(t)$ is the Heaviside step function.

  (ii) Find and sketch the solution of the equation

  $$\frac{d y}{d t}+y=H(t)-H(t-1)$$

  given that $y(0)=0$ and $y(t)$ is continuous.

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• Paper 2, Section II, B

  Differential Equations | Part IA, 2007

  Find the most general solution of the equation

  $$6 \frac{\partial^{2} u}{\partial x^{2}}-5 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=1$$

  by making the change of variables

  $$\xi=x+2 y, \quad \eta=x+3 y .$$

  Find the solution that satisfies $u=0$ and $\partial u / \partial y=x$ when $y=0$.

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• Paper 2, Section II, B

  Differential Equations | Part IA, 2007

  (i) The function $y(z)$ satisfies the equation

  $$y^{\prime \prime}+p(z) y^{\prime}+q(z) y=0$$

  Give the definitions of the terms ordinary point, singular point, and regular singular point for this equation.

  (ii) For the equation

  $$4 z y^{\prime \prime}+2 y^{\prime}+y=0,$$

  classify the point $z=0$ according to the definitions you gave in (i), and find the series solutions about $z=0$. Identify these solutions in closed form.

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Probability

• Paper 2, Section I, F

  Probability | Part IA, 2007

  Let $X$ be a normally distributed random variable with mean 0 and variance 1 . Define, and determine, the moment generating function of $X$. Compute $\mathbb{E} X^{r}$ for $r=0,1,2,3,4$.

  Let $Y$ be a normally distributed random variable with mean $\mu$ and variance $\sigma^{2}$. Determine the moment generating function of $Y$.

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• Paper 2, Section I, F

  Probability | Part IA, 2007

  Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$. Let $U=\min (X, Y)$ and $V=\max (X, Y)$. Show that $\mathbb{E} U=\frac{1}{3}$, and hence find the covariance of $U$ and $V$.

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• Paper 2, Section II, F

  Probability | Part IA, 2007

  Let $A, B$ and $C$ be three random points on a sphere with centre $O$. The positions of $A, B$ and $C$ are independent, and each is uniformly distributed over the surface of the sphere. Calculate the probability density function of the angle $\angle A O B$ formed by the lines $O A$ and $O B$.

  Calculate the probability that all three of the angles $\angle A O B, \angle A O C$ and $\angle B O C$ are acute. [Hint: Condition on the value of the angle $\angle A O B$.]

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• Paper 2, Section II, F

  Probability | Part IA, 2007

  Let $A_{1}, A_{2}, \ldots, A_{n}(n \geqslant 2)$ be events in a sample space. For each of the following statements, either prove the statement or provide a counterexample.

  (i)

  $$P\left(\bigcap_{k=2}^{n} A_{k} \mid A_{1}\right)=\prod_{k=2}^{n} P\left(A_{k} \mid \bigcap_{r=1}^{k-1} A_{r}\right), \quad \text { provided } P\left(\bigcap_{k=1}^{n-1} A_{k}\right)>0$$

  (ii)

  $$\text { If } \sum_{k=1}^{n} P\left(A_{k}\right)>n-1 \text { then } P\left(\bigcap_{k=1}^{n} A_{k}\right)>0$$

  (iii)

  $$\text { If } \sum_{i<j} P\left(A_{i} \cap A_{j}\right)>\left(\begin{array}{c} n \\ 2 \end{array}\right)-1 \text { then } P\left(\bigcap_{k=1}^{n} A_{k}\right)>0 \text {. }$$

  (iv) If $B$ is an event and if, for each $k,\left\{B, A_{k}\right\}$ is a pair of independent events, then $\left\{B, \cup_{k=1}^{n} A_{k}\right\}$ is also a pair of independent events.

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• Paper 2, Section II, F

  Probability | Part IA, 2007

  Let $X$ and $Y$ be independent non-negative random variables, with densities $f$ and $g$ respectively. Find the joint density of $U=X$ and $V=X+a Y$, where $a$ is a positive constant.

  Let $X$ and $Y$ be independent and exponentially distributed random variables, each with density

  $$f(x)=\lambda e^{-\lambda x}, \quad x \geqslant 0$$

  Find the density of $X+\frac{1}{2} Y$. Is it the same as the density of the random variable $\max (X, Y) ?$

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• Paper 2, Section II, F

  Probability | Part IA, 2007

  Let $N$ be a non-negative integer-valued random variable with

  $$P\{N=r\}=p_{r}, \quad r=0,1,2, \ldots$$

  Define $\mathbb{E} N$, and show that

  $$\mathbb{E} N=\sum_{n=1}^{\infty} P\{N \geqslant n\} .$$

  Let $X_{1}, X_{2}, \ldots$ be a sequence of independent and identically distributed continuous random variables. Let the random variable $N$ mark the point at which the sequence stops decreasing: that is, $N \geqslant 2$ is such that

  $$X_{1} \geqslant X_{2} \geqslant \ldots \geqslant X_{N-1}<X_{N},$$

  where, if there is no such finite value of $N$, we set $N=\infty$. Compute $P\{N=r\}$, and show that $P\{N=\infty\}=0$. Determine $\mathbb{E} N$.

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