MATH 310: Applied Mathematics for Science and Engineering

Midterm Review Topics

March 11, 2025

1. Characteristics of differential equations; initial value problems; terminology

2. Population growth models and their variants

3. Graphical methods: direction fields, autonomous equations

4. Solving separable equations

5. Solving first-order linear equations: method of integrating factor; method of variation of parameter

6. Solving exact equations

7. General characteristics of numerical methods; Euler's method

8. Linear models: radioactive decay, carbon dating, law of cooling

9. Non-linear models: logistic growth

10. Theory of higher-order linear equations: existence and uniqueness, linear independence of functions, Wronskian, fundamental set, superposition principles; Theorems 3.1.1 through 3.1.7

11. Method of reduction of order

12. Method of similarity/undetermined coefficients

13. Handling "conflicts" in the method of undetermined coefficients

14. Method of variation of parameters

15. Solving Cauchy-Euler equations

16. Solving differential equations with the Taylor/Maclaurin series method

17. Linear second-order models: mass-spring-damper systems and LRC circuits

Recommended optional review problems:

Chapter 1 Review: 15, 25, 27, 31

Chapter 2 Review: 11, 13, 15, 28, 29, 31a

Chapter 3 Review: 10abcd, 17, 21, 23, 25, 33, 35

Use the Maclaurin series method to solve the IVP: y'' + y' = x; y(0) = 1, y'(0) = 0