Math 330: Ordinary Differential Equations


October 18, 2025


  1. 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

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

  6. Separable equations

  7. Applications: compound interest, population growth, radioactive decay, solution mixing, Newton's Law of cooling

  8. Existence and uniqueness theorem for first order linear equations

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

  10. Bernoulli equation

  11. Logistic model of population growth; carrying capacity

  12. Exact equations

  13. Euler’s numerical method for first order equations

  14. Picard’s method of successive approximations

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

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

  17. The superposition principle (Theorem 3.2.2)

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

  19. Case of two distinct real roots of characteristic equation

  20. Linear differential operator notation

  21. Wronski determinant (Wronskian)

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

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

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

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

  26. 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)

  27. Solving non-homogeneous equations; finding particular solutions using method of similarity (MOS) and method of undetermined coefficients (MOUC); both in straightforward cases as well as cases with conflict(s) between the homogeneous equation's solution and the function g(t)

  28. Solving non-homogeneous equations with the method of variation of parameters