Exam 1 Review
MA 211: Differential Equations I

Chapter 1 Objectives 1. Be able to determine the order of a given ODE and to determine whether it is linear or nonlinear. 2. Be able to determine whether a given family of functions is a solution to a given ODE. 3. Be able to use given initial conditions and a family of solutions to solve an initial value problem. 4. Understand the statement and the implications of the Existence and Uniqueness Theorem. 5. Be able to set up an initial value problem that models a physical process (population, mixtures, Newton’s law of cooling, etc) Chapter 2 Objectives 1. Be able to obtain direction fields for 1st order ODEs and to use them to describe solutions. 2. Be able to find the equilibrium solutions of an autonomous 1st order ODE, to classify the critical points by stability; be able to construct the phase line and to use it to qualitatively describe solutions. 3. Be able to recognize separable 1st order ODEs, to find their solutions, and to solve related initial value problems. 4. Be able to recognize linear 1st order ODEs, to solve them using the method of integrating factors, and to solve related initial value problems. 5. Be able to set up and solve first-order initial value problems for modeling applications involving growth and decay, Newton’s law of cooling, mixture problems, air resistance, and series circuits.

Exam 1 Review problems
1. Determine whether the staement is true or false. If it is true, explain why. If it is false, provide a counterexample or an explanation. a) Every separable differential equation is autonomous. b) Every autonomous differential equation is separable. c) The solution of dy = 4y(y − 1) with y(0) = 2 satisfies y(t) > 1 for all t. dt d) The solution of dy dt

= 4y(y − 1) with y(0) = 2 satisfies limt→∞ = 1.

2. Solve the initial value problems. a) dy 2y − = 2x2 , dx x b) dy = (y 2 + 1)x, y(0) = 1 dx 3. Consider a pond that has a volume of 10,000 cubic meters. Suppose that at time t = 0, the water in the pond is clean and that the pond has two streams flowing into it, stream A and stream B, and one stream flowing out, stream C. Suppose 500 cubic meters per day flow into the pond from stream A, 750 cubic meters per day flow into the pond from stream B, and 1250 cubic meters flow out of the pond via stream C. At time t = 0, the water flowing into the pond from stream A becomes contaminated with road salt at a concentration of 5 kilograms per 1000 cubic meters. Suppose the water in the pond is well mixed. a) Write a differential equation for the amount of salt (in kilograms) in the pond at time t. b) Determine