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MATHEMATICAL TRIPOS Part IA
Friday 28 May 2010 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
Find the general solutions to the following difference equations for yn, n ∈ N.
(i) y (^) n+3 − 3 y (^) n+1 + 2 yn = 0, (ii) y (^) n+3 − 3 y (^) n+1 + 2 yn = 2n, (iii) y (^) n+3 − 3 y (^) n+1 + 2 yn = (−2)n, (iv) y (^) n+3 − 3 y (^) n+1 + 2 yn = (−2)n^ + 2n.
2A Differential Equations
Let f (x, y) = g(u, v) where the variables {x, y} and {u, v} are related by a smooth,
invertible transformation. State the chain rule expressing the derivatives ∂g ∂u
and ∂g ∂v
in
terms of ∂f ∂x and ∂f ∂y and use this to deduce that
∂^2 g ∂u ∂v
∂x ∂u
∂x ∂v
∂^2 f ∂x^2
∂x ∂u
∂y ∂v
∂x ∂v
∂y ∂u
∂^2 f ∂x ∂y
∂y ∂u
∂y ∂v
∂^2 f ∂y^2
+ H
∂f ∂x
+ K
∂f ∂y
where H and K are second-order partial derivatives, to be determined.
Using the transformation x = uv and y = u/v in the above identity, or otherwise, find the general solution of
x ∂^2 f ∂x^2
y^2 x
∂^2 f ∂y^2
∂f ∂x
y x
∂f ∂y
Part IA, Paper 2
SECTION II
5A Differential Equations
(a) Consider the differential equation
an
dny dxn^
dn−^1 y dxn−^1 +... + a 2
d^2 y dx^2
dy dx
with n ∈ N and a 0 ,... , an ∈ R. Show that y(x) = eλx^ is a solution if and only if p(λ) = 0 where p(λ) = anλn^ + an− 1 λn−^1 +... + a 2 λ^2 + a 1 λ + a 0.
Show further that y(x) = xeμx^ is also a solution of (1) if μ is a root of the polynomial p(λ) of multiplicity at least 2.
(b) By considering v(t) =
d^2 u dt^2 , or otherwise, find the general real solution for u(t)
satisfying
d^4 u dt^4
d^2 u dt^2 = 4t^2. (2)
By using a substitution of the form u(t) = y(t^2 ) in (2), or otherwise, find the general real solution for y(x), with x positive, where
4 x^2
d^4 y dx^4
d^3 y dx^3
d^2 y dx^2
dy dx = x.
Part IA, Paper 2
6A Differential Equations (a) By using a power series of the form
y(x) =
∑^ ∞
k=
ak xk
or otherwise, find the general solution of the differential equation
xy′′^ − (1 − x)y′^ − y = 0. (1)
(b) Define the Wronskian W (x) for a second order linear differential equation
y′′^ + p(x)y′^ + q(x)y = 0 (2)
and show that W ′^ + p(x)W = 0. Given a non-trivial solution y 1 (x) of (2) show that W (x) can be used to find a second solution y 2 (x) of (2) and give an expression for y 2 (x) in the form of an integral.
(c) Consider the equation (2) with
p(x) = − P (x) x
and q(x) = − Q(x) x where P and Q have Taylor expansions
P (x) = P 0 + P 1 x +... , Q(x) = Q 0 + Q 1 x +...
with P 0 a positive integer. Find the roots of the indicial equation for (2) with these assumptions. If y 1 (x) = 1 + βx +... is a solution, use the method of part (b) to find the first two terms in a power series expansion of a linearly independent solution y 2 (x), expressing the coefficients in terms of P 0 , P 1 and β.
Part IA, Paper 2 [TURN OVER
9F Probability (a) What does it mean to say that a random variable X with values n = 1, 2 ,... has a geometric distribution with a parameter p where p ∈ (0, 1)? An expedition is sent to the Himalayas with the objective of catching a pair of wild yaks for breeding. Assume yaks are loners and roam about the Himalayas at random. The probability p ∈ (0, 1) that a given trapped yak is male is independent of prior outcomes. Let N be the number of yaks that must be caught until a breeding pair is obtained. (b) Find the expected value of N. (c) Find the variance of N.
10F Probability The yearly levels of water in the river Camse are independent random variables X 1 , X 2 ,.. ., with a given continuous distribution function F (x) = P(Xi 6 x), x > 0 and F (0) = 0. The levels have been observed in years 1,.. ., n and their values X 1 ,.. ., Xn recorded. The local council has decided to construct a dam of height
Yn = max
[
X 1 ,... , Xn
]
Let τ be the subsequent time that elapses before the dam overflows:
τ = min
[
t > 1 : Xn+t > Yn
]
(a) Find the distribution function P(Yn 6 z), z > 0, and show that the mean value EYn =
0
[1 − F (z)n]dz.
(b) Express the conditional probability P(τ = k | Yn = z), where k = 1, 2 ,... and z > 0, in terms of F. (c) Show that the unconditional probability
P(τ = k) = n (k + n − 1)(k + n)
, k = 1, 2 ,....
(d) Determine the mean value E τ.
Part IA, Paper 2 [TURN OVER
11F Probability
In a branching process every individual has probability pk of producing exactly k offspring, k = 0 , 1 ,.. ., and the individuals of each generation produce offspring independently of each other and of individuals in preceding generations. Let Xn represent the size of the nth generation. Assume that X 0 = 1 and p 0 > 0 and let Fn(s) be the generating function of Xn. Thus
F 1 (s) = EsX^1 =
∑^ ∞
k=
pksk, |s| 6 1.
(a) Prove that Fn+1(s) = Fn(F 1 (s)).
(b) State a result in terms of F 1 (s) about the probability of eventual extinction. [No proofs are required.]
(c) Suppose the probability that an individual leaves k descendants in the next generation is pk = 1/ 2 k+1, for k > 0. Show from the result you state in (b) that extinction is certain. Prove further that in this case
Fn(s) =
n − (n − 1)s (n + 1) − ns , n > 1 ,
and deduce the probability that the nth generation is empty.
Part IA, Paper 2
