Mathematics 304 – Ordinary Differential Equations Review Sheet – Exam 1 September 28, 2000

General Information

The first midterm exam this semester will be given in class on Thursday, October 5, as announced in the course syllabus. This will be a closed-book exam, but you may use calculators at any point, and I will provide a copy of the table of integrals from the endpapers of our textbook for your use during the exam.

The exam will cover the material from Chapters 1 - 3 of the text that we have studied since the start of the semester (see specific topics and review problems below).

If there is interest, I would be happy to arrange a late afternoon or evening review session for the exam. Tuesday, October 3 is probably the best day to aim for.

Topics To Be Included

1) ODE, solutions, initial value problems, the Existence and Uniqueness Theorem for solutions of first order initial value problems

2) Qualitative theory for 1st order ODE: Direction Fields, special techniques for autonomous equations (identifying equilibrium solutions, determining whether they are “stable” or “unstable” – see section 1.4 in the text and Discussion 1)

3) Techniques for deriving analytic solutions of 1st order ODE: Separable equations, linear equations, exact equations, integrating factors, substitutions and transformations for homogeneous equations, equations of the form \frac{dy}{dx} = g(ax + by + c), Bernoulli equations.

4) 1st order ODE in modeling of mixing problems, heating/cooling, etc.

Suggested Review Problems

From §1.2: 16, 17, 18, 20, 21, 23 - 28

From §1.3: 1, 3, 4, 5

From §1.4: 3, 4, 5 (Note: a “sink” is an equilibrium with the property that all solutions that start with y near the equilibrium level tend toward that level as the independent variable tends to +∞; a “source” is an equilibrium with the property that all solutions that start with y near the equilibrium level tend away from that level as the independent variable tends to +∞. Equilibria that are neither sinks nor sources are called “nodes”.)

From Review Problems at end of Chapter 2 (page 87): try a good selection of these. Of course the first issue is to decide which class of equation you have and what method(s) apply.

From §3.2: 1, 3, 7, 9 (the “Malthusian law” is \frac{dP}{dt} = kP for some constant k), 13, 18 From §3.3: 1, 3, 7, 11 (Note: in the ODE

\frac{dT}{dt} = K(A(t) − T) + H(t) + K_{U}(T_{D} − T)

representing “proportional control” heating and air conditioning (T_{D} = desired temperature, H(t) is the heat added from sources other than the heating/ac system, A(t) is the ambient temperature), the “time constant” with heating/ac is \frac{1}{K+K_{U}}; the time constant for the building or van without heating/ac is \frac{1}{K}.