# Test 1 Review — Math 308-502 (Summer 2015)

## General Information

Test 1 will be given on Friday, 7/3/15, during our usual class time and in our usual classroom. I will have extra office hours on Thursday afternoon, 1-4 pm, and on Friday morning, 9:30-10:30 am.

Bluebooks. Please bring an 8½x11 bluebook.

Calculators. You may use scientific calculators to do numerical calculations — logs, exponentials, and so on. You may not use any calculator that has the capability of doing algebra or calculus, or of storing course material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Structure and Coverage There will be 6 to 8 questions, some with multiple parts. The problems will be similar to homework problems, examples done in class and worked out in the text. The material covered is from the following sections in the text: 1.2, 2.1-2.6, 3.1-3.7.

## Topics covered

### First Order ODEs

• Methods
• Integrating factors— Be able to find and use integrating factors to solve 1st order linear ODEs, including initial value problems.
• Separable equations— Be able to determine when an equation is separable and know how to solve the equation, and able to solve initial value problems.
• Nonlinear vs. Linear equations— Elementary requirements for most physical models: A solution must exist and there must be only one solution. Linear equations have these properties; however, nonlinear ones may not.
• Exact equations — Be able to determine whether or not an equation is exact. Be able to solve exact equations, including initial value problems.

• Applied Problems
• Mixing problems — Set these up using dx/dt = rate in − rate out. Be sure to check units!
• Newton's law of cooling — Temperature in a building and "time-of-death" problems
• Circuits — Be able to set up and solve simple circuits.
• Falling bodies & rockets — Air friction is modeled by friction force = −bv. Be able to find limiting velocities (section 1.2). Be able to find the escape velocity of a rocket.
• Population dynamics. — Logistic model. Gompertz model (problem 2.5.16). Phase line. Equilibrium points. Stability of an equilibrium point.

### Second Order ODEs

• Foundations
• Linear equations, initial value problems, operator notation
• Homogeneous linear equations: principle of superposition, Wronskians
• Wronskians, fundamental sets, Abel's theorem. (Be able to state and prove this.)
• Nonhomogeneous equations: Any two solutions differ by a homogeneous solution
• Superposition of forces

• Methods
• Constant coefficient homogeneous equations: Find characteristic polynomial. Three cases: two distinct real roots; one double real root; complex roots. Be able to solve initial value problems. (Sections 3.1, 3.3)
• Reduction of order. Use this to find a second solution for the fundamental set. (Section 3.2)
• Undetermined coefficients. The table on p. 181 will be given.
• Variation of parameters. Formula (26), p. 188 will be given.

• Applied problems.
• Be able to solve initial value problems.
• Be able to set up and solve free vibration problems -- springs and circuits. Terms you should know and be able to find: undamped, natural frequency and period, damped frequency and period, amplitude and phase. (Section 3.7)
Updated 6/29/2015.