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EGR 265, Math Tools for Engineering Problem Solving
September 15, 2008, 50 minutes
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TEST I
Problem 1
Determine the order of the following ODEs. Also, state if they are linear or non-linear. (4P+4P+4P)
(a) yy′′^ = cos x
(b) y^4 − y′′′^ = exy
(c)
y − cos x y′^
= ex
Problem 2
Which of the functions y = 0, y = 2, y = 2x and y = 2x^2 is a solution of the following DEs? (5P+5P+5P)
(a) xy′^ = y
(b) y′′^ + 9y = 18
(c) xy′′^ − y′^ = 0
Note: The DEs (a), (b) and (c) may have more than one solution. Behind each of them list all correct solutions.
Problem 3
Below the direction field for y′^ = x^2 y is given.
(a) Sketch the solution y(x) of y′^ = x^2 y with initial value y(0) = 1. (6P)
(b) From your sketch determine limx→∞ y(x) and limx→−∞ y(x). (4P)
Problem 5
The percentage p(t) of a population which uses cell phones is described by the logistic DE p′^ = p(1 − p), where t is measured in years. Initially, 10 percent of the population use cell phones, i.e. p(0) = 0.1. Solve the DE to find a formula which gives p(t) for arbitrary times t. Note that, as p(t) is a percentage, we will always have 0 < p(t) < 1. (17P)
Problem 6
In the electrical circuit below one has R = 1000 ohms, C = 0.001 farads and a constant electromotive force of E(t) = 50 volts.
(a) Write down the DE for the charge q(t) and solve it to find a closed form solution for q(t) in Coulombs given that q(0) = 0. (13P)
(b) Find a formula for the current i(t) in amperes. (4P)
