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美国数学代写 Coursework submission deadline is 3pm, Friday 29th Nov. Please ensure that you include your personalised coursework coversheet with your ···
美国数学代写
Coursework submission deadline is 3pm, Friday 29th Nov. Please ensure that you include your personalised coursework coversheet with your submission. These can be obtained from the coversheet portal on Moodle.
1.The figure below visually captures all of this information.Two connected tanks (tank X and tank Y) are to be used to mix a chemical with water. Each
tank holds 20 litres of pure water at time ? = 0. The chemical is added continuously to tank X at a rate of 1 l/min. Pure water is added continuously to tank Y at a rate of 2 l/min. Pumps exchange solutions between the tanks. The solution from tank X is pumped into tank Y at 2 l/min. The solution from tank Y is pumped into tank X at 1 l/min and out of the system at 3 l/min.
The figure below visually captures all of this information.
Let ?(?) and ?(?) denote the concentrations of the chemical in tanks X and Y respectively over time.美国数学代写
This set-up can be described by the following system of ODEs:
(a) By eliminating ? in equation (1), show that
(b) Hence (or otherwise), obtain the unique solution for ?(?), giving your answer in exact form (i.e. don’t use rounded decimals) and in the simplest form. You do not need to find ?(?).
Hint: When applying the initial conditions to fix your constants of integration, the algebra is somewhat easier if we consider ?′ (0) rather than ?(0). You can obtain the initial condition for ?′ (0) by evaluating equation (1) at ? = 0.
2.An evil genius has a secret lab up in the Arctic.美国数学代写
However, his secret lab is not particularly well insulated. To protect his lab equipment (and his geraniums) from the cold, the heating automatically comes on, at a constant intensity, every other hour. We can say that:
Rate of change of Increase from Loss of heat to
= −
heat inside lab heating colder surroundings
A simple ODE model that portrays these dynamics is the following:
d?
— =30? (?) − ? ,
d?
where ? (?) denotes the temperature (in degrees Celsius) inside the lab (we assume the temperature is uniform everywhere inside the lab for simplicity), and
? (?) = { 0 0 ≤ ? < 1,
1 1 ≤ ? < 2, with ? (? + 2) = ? (?).
(a) Show that the Fourier series for ? (?) is given by: 美国数学代写
(b) Given that the temperature inside the lab at time ? = 0 is 25∘?, show that the unique solution to equation (3) is:
Pro-gamer move (another hint): Why not substitute ? = (2? − 1) inside your sum to make the algebra easier and neater? This is a practice that is commonly used by mathematicians and engineers alike to make their lives easier! Just don’t forget to swap back at the end!
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