ODEs: We describe basic properties of the complex numbers. A method for dividing complex numbers is given. Real and imaginary parts of a complex number are defined. Euler's formula is stated, and various consequences are interpreted.

1. General Solution of y' + xy = 0
2. Verifying the Solution of an ODE
3. The Logistic Function 1: Solving The ODE
4. The Logistic Function 2: Sketching The S-Curve
5. General Solution to y' - 3y = b(x)
6. Example of Population Growth 1
7. Example of Population Growth 2
8. Example of Radioactive Decay 1
9. Example of Radioactive Decay 2
10. General Solution to y" - 6y' + 9y = 0
11. Complex Numbers for ODEs (1 of 4)
12. Complex Numbers for ODEs (2 of 4)
13. Complex Numbers for ODEs (3 of 4)
14. Complex Numbers for ODEs (4 of 4)
15. General Solution to y" - 6y' +25y = 0
16. Antiderivative of e^{3x} cos(4x) (ODE Solution)
17. Antiderivative of x^2 e^x (ODE Solution)
18. General Solution of y'''-4y''+5y'-2y=0
19. Wronskian for {e^{3x}, e^{-x}, 2}
20. Linear Dependence of {x^2-1, x^2+x, x+1} Using Wronskian
21. Annihilator Method 1: Real Linear Factors
22. Example of Annihilator Method: y"-y = sin(2x)
23. Power Series Solution for y"-2y'+y=x, y(0)=0, y'(0)=1
24. Mass-Spring Systems 1: Undamped Motion
25. Mass-Spring Systems 2: Underdamped Motion
26. Mass-Spring Systems 3: Critically Damped Motion
27. Laplace Transform of f(t) = 2t-1
28. Laplace Transform of f(t) = sin(2t)
29. Laplace Transform of f(t) = t sin(2t)
30. Laplace Transform of f(t) = e^{3t}cos(4t)
31. Laplace Transform of f(t) = t^2 e^{2t} cos(3t)
32. Inverse Laplace Transform of (s-1)/s^2(s^2+4)
33. Laplace Transform Solution of y'-3y=e^{2t}, y(0)=2
34. Laplace Transform Solution of y"-2y'-3y=e^t, y(0) = 0, y'(0) = 1
35. Laplace Transform of f(t) = 2 on the Interval (1,2)
36. Second Shift Formula for a Piecewise-defined Function
37. Laplace Transform Solution of y'-y=f(t) (Piecewise-Defined)
38. Example of Convolution Theorem: f(t)=t, g(t)=sin(t)
39. Convolution Theorem for y'-2y=e^t, y(0)=0
40. Fourier Series: Example of Orthonormal Set of Functions
41. Fourier Series: Example of Parseval's Identity

Dr. Bob explains ordinary differential equations, offering various examples of first and second order equations, higher order differential equations using the Wronskian determinant, Laplace transforms, and more.