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MATHEMATICAL TRIPOS Part IA
Friday, 3 June, 2011 1:30 pm to 4:30 pm
PAPER 2
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt all four questions from Section I and at most five questions from Section II. In Section II, no more than three questions on each course may be attempted.
Complete answers are preferred to fragments.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in separate bundles, marked A, B, C, D, E and F according to the code letter affixed to each question. Include in the same bundle all questions from Section I and II with the same code letter.
Attach a completed gold cover sheet to each bundle.
You must also complete a green master cover sheet listing all the questions you have attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Gold cover sheets None Green master cover sheet
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
SECTION I
1A Differential Equations
(a) Consider the homogeneous kth-order difference equation
akyn+k + ak− 1 yn+k− 1 +... + a 2 yn+2 + a 1 yn+1 + a 0 yn = 0 (∗)
where the coefficients ak,... , a 0 are constants. Show that for λ 6 = 0 the sequence yn = λn is a solution if and only if p(λ) = 0, where
p(λ) = akλk^ + ak− 1 λk−^1 +... + a 2 λ^2 + a 1 λ + a 0.
State the general solution of (∗) if k = 3 and p(λ) = (λ − μ)^3 for some constant μ.
(b) Find an inhomogeneous difference equation that has the general solution
yn = a 2 n^ − n , a ∈ R.
2A Differential Equations
(a) For a differential equation of the form d dxy = f (y), explain how f ′(y) can be used to determine the stability of any equilibrium solutions and justify your answer.
(b) Find the equilibrium solutions of the differential equation
dy dx = y^3 − y^2 − 2 y
and determine their stability. Sketch representative solution curves in the (x, y)-plane.
3F Probability
Let X be a random variable taking non-negative integer values and let Y be a random variable taking real values.
(a) Define the probability-generating function GX (s). Calculate it explicitly for a Poisson random variable with mean λ > 0.
(b) Define the moment-generating function MY (t). Calculate it explicitly for a normal random variable N(0, 1).
(c) By considering a random sum of independent copies of Y , prove that, for general X and Y , GX
MY (t)
is the moment-generating function of some random variable.
Part IA, Paper 2
SECTION II
5A Differential Equations
(a) Find the general real solution of the system of first-order differential equations
x˙ = x + μy y ˙ = −μx + y ,
where μ is a real constant.
(b) Find the fixed points of the non-linear system of first-order differential equations
x˙ = x + y y ˙ = −x + y − 2 x^2 y
and determine their nature. Sketch the phase portrait indicating the direction of motion along trajectories.
6A Differential Equations
(a) A surface in R^3 is defined by the equation f (x, y, z) = c, where c is a constant. Show that the partial derivatives on this surface satisfy
∂x ∂y
z
∂y ∂z
x
∂z ∂x
y
(b) Now let f (x, y, z) = x^2 − y^4 + 2ay^2 + z^2 , where a is a constant.
(i) Find expressions for the three partial derivatives ∂x ∂y
z
, ∂y ∂z
x
and ∂z ∂x
y
on the surface f (x, y, z) = c, and verify the identity (∗). (ii) Find the rate of change of f in the radial direction at the point (x, 0 , z). (iii) Find and classify the stationary points of f. (iv) Sketch contour plots of f in the (x, y)-plane for the cases a = 1 and a = −1.
Part IA, Paper 2
7A Differential Equations (a) Define the Wronskian W of two solutions y 1 (x) and y 2 (x) of the differential equation y′′^ + p(x)y′^ + q(x)y = 0 , (∗) and state a necessary and sufficient condition for y 1 (x) and y 2 (x) to be linearly independ- ent. Show that W (x) satisfies the differential equation
W ′(x) = −p(x)W (x).
(b) By evaluating the Wronskian, or otherwise, find functions p(x) and q(x) such that (∗) has solutions y 1 (x) = 1 + cos x and y 2 (x) = sin x. What is the value of W (π)? Is there a unique solution to the differential equation for 0 6 x < ∞ with initial conditions y(0) = 0, y′(0) = 1? Why or why not? (c) Write down a third-order differential equation with constant coefficients, such that y 1 (x) = 1 + cos x and y 2 (x) = sin x are both solutions. Is the solution to this equation for 0 6 x < ∞ with initial conditions y(0) = y′′(0) = 0, y′(0) = 1 unique? Why or why not?
8A Differential Equations (a) The circumference y of an ellipse with semi-axes 1 and x is given by
y(x) =
∫ (^2) π
0
sin^2 θ + x^2 cos^2 θ dθ. (∗)
Setting t = 1 − x^2 , find the first three terms in a series expansion of (∗) around t = 0. (b) Euler proved that y also satisfies the differential equation
x(1 − x^2 )y′′^ − (1 + x^2 )y′^ + xy = 0.
Use the substitution t = 1 − x^2 for x > 0 to find a differential equation for u(t), where u(t) = y(x). Show that this differential equation has regular singular points at t = 0 and t = 1. Show that the indicial equation at t = 0 has a repeated root, and find the recurrence relation for the coefficients of the corresponding power-series solution. State the form of a second, independent solution. Verify that the power-series solution is consistent with your answer in (a).
Part IA, Paper 2 [TURN OVER
11F Probability I was given a clockwork orange for my birthday. Initially, I place it at the centre of my dining table, which happens to be exactly 20 units long. One minute after I place it on the table it moves one unit towards the left end of the table or one unit towards the right, each with probability 1/2. It continues in this manner at one minute intervals, with the direction of each move being independent of what has gone before, until it reaches either end of the table where it promptly falls off. If it falls off the left end it will break my Ming vase. If it falls off the right end it will land in a bucket of sand leaving the vase intact. (a) Derive the difference equation for the probability that the Ming vase will survive, in terms of the current distance k from the orange to the left end, where k = 1,... , 19. (b) Derive the corresponding difference equation for the expected time when the orange falls off the table. (c) Write down the general formula for the solution of each of the difference equations from (a) and (b). [No proof is required.] (d) Based on parts (a)–(c), calculate the probability that the Ming vase will survive if, instead of placing the orange at the centre of the table, I place it initially 3 units from the right end of the table. Calculate the expected time until the orange falls off. (e) Suppose I place the orange 3 units from the left end of the table. Calculate the probability that the orange will fall off the right end before it reaches a distance 1 unit from the left end of the table.
12F Probability A circular island has a volcano at its central point. During an eruption, lava flows from the mouth of the volcano and covers a sector with random angle Φ (measured in radians), whose line of symmetry makes a random angle Θ with some fixed compass bearing. The variables Θ and Φ are independent. The probability density function of Θ is constant on (0, 2 π) and the probability density function of Φ is of the form A(π − φ/2) where 0 < φ < 2 π, and A is a constant. (a) Find the value of A. Calculate the expected value and the variance of the sector angle Φ. Explain briefly how you would simulate the random variable Φ using a uniformly distributed random variable U. (b) H 1 and H 2 are two houses on the island which are collinear with the mouth of the volcano, but on different sides of it. Find (i) the probability that H 1 is hit by the lava; (ii) the probability that both H 1 and H 2 are hit by the lava; (iii) the probability that H 2 is not hit by the lava given that H 1 is hit.
Part IA, Paper 2 [TURN OVER
END OF PAPER
Part IA, Paper 2
