MATH 310: Applied Mathematics for
Science and Engineering
Final Exam Review Topics
May 19, 2025
1. Characteristics of differential equations; 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, RL and RC circuits
9. Theory of higher-order linear equations: existence and
uniqueness, linear independence of functions, Wronskian, fundamental set,
superposition principles
10. Method of reduction of order
11. Method of similarity/undetermined coefficients;
handling "conflicts"
12. Method of variation of parameters
13. Solving Cauchy-Euler equations
14. Solving differential equations with the Taylor series
method
15. Linear models: mechanical vibrations; electrical
circuit oscillations
16. Laplace transform; definition; existence
17. Laplace transform for piecewise-defined functions
18. Laplace transforms of
derivatives
19. Solving initial value problems with Laplace transform
20. First and Second Translation Theorems for Laplace
transform
21. Using the unit-step function and Dirac's Delta in
solving initial value problems
22. Solving systems of differential equations with Laplace
transform
23. Power series solutions of differential equations
24. Vectors in 2D and 3D: properties, operations, dot and
cross product
25. Matrices: properties, operations
26. Row operations; Gauss elimination; solving linear
systems
27. Rank of a matrix
28. Determinants and their properties
29. Inverse matrix: two methods of finding the inverse
30. Using inverse matrix to solve linear systems
31. Solving systems with Cramer's Rule
32. Eigenvalues and eigenvectors of a matrix
33. Complex eigenvalues and eigenvectors
34. Powers of matrices
35. Symmetric and orthogonal matrices
36. Diagonalization
37. The method of least squares
Recommended optional review problems:
Chapter 1 Review: 29, 33
Chapter 2 Review: 9, 15, 28,
29, 31a
Chapter 3 Review: 9, 19, 23,
27, 33, 37
Chapter 4 Review: 17, 19, 29,
35, 37, 41
Chapter 5 Review: 11, 13, 15
Chapter 7 Review: 20, 23, 27,
30, 31
Chapter 8 Review: 25, 27, 35,
39, 41, 43, 45, 47, 59