Math 330: Ordinary Differential Equations

October 16, 2015

Midterm Exam Review Topics

1. Direction fields for first order equations

2. Autonomous equations; equilibrium solutions; asymptotically stable, unstable and semi-stable solutions

3. First order equations; initial value problems

4. First order linear equations; the method of integrating factor – derivation and use

5. First order linear equations; the method of variation of parameter

6. Separable equations

7. Models of mixing process

8. Continuous compounding of interest

9. Simple population growth models; exponential growth

10. Radioactive decay model

11. Existence and uniqueness theorem for first order linear equations

12. Existence and uniqueness theorem for non-linear first order equations

13. Bernoulli equation

14. Logistic model of population growth; carrying capacity

15. Exact equations

16. Theorem about conditions for exactness

17. Euler’s numerical method for first order equations; components of error of a numerical method

18. Picard’s method of successive approximations

19. Second order linear differential equations; initial value problems; homogeneous and non-homogeneous equations

20. The superposition principle (Theorem 3.2.2)

21. Existence and uniqueness theorem for second-order linear differential equations (Theorem 3.2.1)

22. Second-order linear homogeneous equations with constant coefficients; characteristic equation

23. Case of two distinct real roots of characteristic equation

24. Linear differential operator notation

25. Wronski determinant (Wronskian)

26. Theorems about constants (TACO) (Theorem 3.2.3) and about the general solution (TAGS) (Theorem 3.2.4); fundamental set of solutions

27. Theorem about complex-valued solution of a second order linear equation (Theorem 3.2.6)

28. Abel’s theorem (AT) (Theorem 3.2.7)

29. Case of complex roots of characteristic equation; Euler’s formula; obtaining real-valued solutions

30. Case of repeated roots of characteristic equation; method of reduction of order

31. Second-order equations reducible to first-order equations: two types – dependent variable missing and independent variable missing (see pages 135-136)

32. Non-homogeneous second order linear differential equations; DIFIFI Theorem (Theorem 3.5.1) and Theorem about the general solution of a non-homogeneous linear equation (Theorem 3.5.2)

33. Solving non-homogeneous equations; finding particular solutions using method of similarity (MOS) and method of undetermined coefficients (MOUC)

34. Superposition principle for non-homogeneous equations (stated on the lower half of page 179)

35. The method of variation of parameters (PARVAR)