Reference no: EM132321229

Mathematics Questions -

Q1. A climate scientist makes the following observations on the weather in Ballarat in winter:

i. There are never two consecutive days of nice weather.

ii. Following a nice day, there are equal probabilities (i.e. 1/2) of having either a rainy day or a windy day.

iii. Following a rainy day or a windy day, the probability of having the same weather the next day is 1/2, and the probability of having a nice day is 1/4.

a) What is the state space of this process?

b) Write down the Markov chain modelling the weather in Ballarat according to these observations.

c) Show that this is a regular Markov chain.

d) Find the equilibrium (stationary) distribution of the Markov chain.

e) In the 3 months (90 days) of winter, how many days of nice weather can be expected?

Q2. Let u(x, t) be a function of two variables which satisfies the following partial differential equation:

∂u/∂x + u/x = 3x

a) Solve to determine u. (Note that the PDE contains only u, derivatives of u, and functions of x.)

b) Find the solution which satisfies u(1, t) = sin²t.

c) Find any non-constant solution to the following partial differential equation:

∂²u/∂x∂y + 1/x ∂u/∂y = 0

Q3. Rewrite the third-order differential equation

cos ty'''(t) + √ty'(t) - e^ty(t) = 2t

as a system of first-order differential equation.

Note - Needing all work shown. And answers to be correct.