MATH 280, Study Guide for Midterm Test 1

7/22/2014

Test coverage

1. Introduction to Differential Equations (Chapter 1)
   1. Basic Examples. Solutions of Differential Equations. Interval of Existence (1.1)
   2. Initial Value Problems. Existence-Uniqueness Theorem (1.2)
   3. Differential Equations and Mathematical Models (1.3)
2. First-Order Differential Equations (Chapter 2)
   1. Solutions Curves without a Solution (2.1)
   2. Separable equations (2.2)
   3. Linear equations (2.3)
   4. Exact Equations. Integrating factors (2.4)
   5. Solutions by substitution (2.5)
   6. A Numerical Method (2.6)
3. Modeling with First-Order Differential Equations (Chapter 3)
   1. Linear Models (3.1)
   2. Nonlinear Models (3.2)
   3. Modeling with Systems of First-Order ODE (3.3)

Key concepts (find in the textbook/lecture notes -- know for the test)

• Differential equation, solution, interval of existence
• Conditions of existence-uniqueness theorem, examples of non-uniqueness
• Family of solutions, general solution, singular solution, initial data
• Direction field, autonomous and non-autonomous equations; isoclines
• Singular points, phase portrait, solution curves
• Equilibrium solutions, stability/instability of equilibrium solutions
• Basic types of solvable first-order equations: separable, linear, exact, homogeneous, Bernoulli...
• Solution by substitution
• Euler's numerical method
• Population models: unlimited growth, logistic (limited growth), "doomsday equation"
• Mixing models
• Newton's law of cooling
• Radioactive decay

Review questions:

1.Rev: 17-19, 22, 23, 25-27, 38; 2.Rev: 15, 18, 27-29, 30, 32; 3.Rev: 4, 11, 12.

Good luck on the exam!