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An exam paper for the course Differential Equations offered by the Mathematical Sciences Department at Carnegie Mellon University. The exam consists of four questions, each worth 24 points, and covers topics such as solving initial value problems and modeling population growth using differential equations. The exam does not allow the use of calculators and requires students to show all work and provide clear explanations.
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Mathematical Sciences Department Differential Equations Carnegie Mellon University 21-
Fall 2007 Exam 1
No calculator of any kind is permitted. Show all work and give clear explanations.
NAME:
Question Points Score Pres Pt
1 24+
Total 100
(a) Solve the initial value problem y′^ = (1 − y)^2 sin t y(0) = 1 / 2 and determine the maximal domain (aka “interval of validity”) of the solution. (b) If the initial condition is changed to y(0) = 0, what is the maximal domain?
(a) Determine conditions on (t 0 , y 0 ) for which the initial value problem
y′^ = ln(y − 1 /|t|)^2 y(t 0 ) = y 0
is guaranteed to have a unique local solution. (b) Could there be any solutions to y′^ = ln(y − 1 /|t|)^2 which are defined on the whole real line, (−∞, ∞)? Explain.
(a) Write down a differential equation that models population in the lake. (b) Suppose the lake is opened to recreational fishing. Assume that the more populous are the fish, the easier they are to catch, and the less populous, the harder to catch. Write down a differential equation that models this scenario. (c) There must be some proportionality constant in your equation in (b). Would it be reasonable to expect that this constant could be greater than 0.01? Explain. (d) Draw a clearly labeled solution portrait corresponding to your equation in (b). Consider only nonnegative t and P. (An extra page is provided if needed.)
