Download Math 2930, Fall 2015, Prelim 1 Tuesday, October 6, 7:30-9

Math 2930, Fall 2015, Prelim 1
Tuesday, October 6, 7:30-9:00 pm
This exam is closed book and closed notes. No calculators. Please turn off all cell phones
and other communication devices.
Academic integrity is expected of all students at all time whether in the presence or absence
of members of the teaching staff. If you understand this and agree that you will neither give,
nor use, nor receive unauthorized aid in this examination, please sign the Academic Integrity
space on the examination booklet. Thank you.
1. 20 pts Find the analytical solutions to the following equations
10 pts a) y 0 + ay = be−λt . Note that a, b, λ are constants.
10 pts b)
dy
dx
= a(y − 2)(y − 1), with y > 2. a is a constant.
2. 20 pts Consider the equation
dy
= y(1 − y/2)(1 − y)2
dt
(1)
12 pts
a) Find the equilibrium points and determine their stability.
8 pts
b) Sketch the solutions for three different initial conditions, y(0) = 0.5, y(0) = 1.5,
and y(0) = 4. Please put all three curves on the same plot.
3. 20 pts Suppose you wish to take a loan from a bank which charges an annual interest of r.
The interest is compounded continuously. You can only afford a monthly payment of
$k.
8 pts
a) Write down the differential equation governing the amount of money you owe,
S(t), where t is the time in years.
12 pts b) If you would like to pay off everything in 5 years, how much money, S0 , can you
afford to borrow? Express your answer in terms of r and k.
please turn over
1
4. 20 pts Bacteria population
a) Suppose that a bacterium replicates itself in discrete steps once per second, estimate how long would it take to reach a population of one million, starting with
a population of one? Give an integer answer closest to the exact answer. You do
not need to evaluate log10
2 , and no need to solve a differential equation for this
part.
5 pts
9 pts
b) If we express the population dynamics in terms of the differential equation,
dy
= ry, find the growth rate, r.
dt
c) In order to control the bacteria population, how would you modify the growth rate
so that the population eventually approaches y = 16? Write down the expression
for the growth rate.
6 pts
5. 20 pts Consider the equation
dy
= y
dt
y(0) = 1
(2)
(3)
5 pts a) Find the analytical solution for y(t) for the given initial condition
5 pts b) If instead we solve the equation using the forward Euler’s method, with a step
size h, write down the first 2 iterations. Express you answers in terms of h.
5 pts
c) Based on b), write down the expression after nth iteration.
5 pts d) Let n be the number of steps over the interval [0,t], with n = t/h, show that in
the limit as h → 0, and n → ∞, the numerical answer given by Euler’s method
converges to the analytical solution that you found in part a).
2