Mathematics 304, Section 2 – Ordinary Differential Equations Midterm Exam 1 – October 1, 2004

Directions: Do all work in the blue exam booklet. You may use a calculator and the table of integrals at any point. There are 100 total regular points and 10 Extra Credit points.

I. Consider the first order ODE

(1)  y′ = 2t(y − 2)^{2/3}

A) (15) Find a solution of this equation that satisfies the initial condition y(0) = 3. B) (10) What does the Existence and Uniqueness Theorem say about solutions of the

initial value problem for (1) with y(0) = 2? Show that the constant function y(t) = 2 for all t is one solution. Is it the only one?

II. (25) A tank with capacity 200 gal. originally contains 50 gal. water with 30 lb of salt in solution. Water containing .5 lb of salt per gallon is entering at 3 gal./minute and the tank is thoroughly mixed so the salt concentration is uniform at all times. Contents of the tank are also flowing out of an exit pipe at 2 gal/minute. Find the amount of salt in the tank at all times prior to the time the tank overflows. What is the amount of salt at the instant the tank starts to overflow?

III. Let T, M be positive constants with T < M, and consider the autonomous 1st order ODE

(2)  P′ = −P (1 − \frac{P}{T} ) (1 − \frac{P}{M} )

A) (15) What are the equilibrium solutions of this equation? Which of the equilibria are

sinks and which are sources? B) (10) Suppose the equation (2) models growth of a population of a species of organisms

in a particular environment. Based on your answer to part A, what happens to solutions of initial value problems for (1) if P (0) < T? if T < P (0) < M? if P (0) > M? What are the meanings of the P values P = T and P = M?

IV. (25) Sketch the bifurcation diagram for the family of ODE

y′ = y^{3} − 3ay

containing the parameter a. Describe all bifurcations that occur.

Extra Credit

(10) The ODE y′ = a(t)y + b(t) has solutions y = t^{2} + e^{−t} and y = 3t^{2} + e^{−t}. What are the functions a(t) and b(t)? Is there only one possible answer?