# Ordinary Differential Equations: Solutions and Examples

### Course Description

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.

Dr. Bob explains the Wronskian determinant in Lecture 19: Wronskian for {e^{3x}, e^{-x}, 2}
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### Video Lectures & Study Materials

# Lecture Play Lecture
I. First-Order ODEs
1 General Solution of y' + xy = 0 (7:45) Play Video
2 Verifying the Solution of an ODE (6:06) Play Video
3 The Logistic Function 1: Solving The ODE (14:39) Play Video
4 The Logistic Function 2: Sketching The S-Curve (10:04) Play Video
5 General Solution to y' - 3y = b(x) (7:33) Play Video
6 Example of Population Growth 1 (4:51) Play Video
7 Example of Population Growth 2 (5:49) Play Video
8 Example of Radioactive Decay 1 (5:03) Play Video
9 Example of Radioactive Decay 2 (7:23) Play Video
II. Second-Order ODEs
10 General Solution to y" - 6y' + 9y = 0 (3:21) Play Video
11 Complex Numbers for ODEs (1 of 4) (14:42) Play Video
12 Complex Numbers for ODEs (2 of 4) (11:51) Play Video
13 Complex Numbers for ODEs (3 of 4) (12:51) Play Video
14 Complex Numbers for ODEs (4 of 4) (4:28) Play Video
15 General Solution to y" - 6y' +25y = 0 (5:45) Play Video
16 Antiderivative of e^{3x} cos(4x) (ODE Solution) (4:19) Play Video
17 Antiderivative of x^2 e^x (ODE Solution) (3:20) Play Video
18 General Solution of y'''-4y''+5y'-2y=0 (6:24) Play Video
19 Wronskian for {e^{3x}, e^{-x}, 2} (8:23) Play Video
20 Linear Dependence of {x^2-1, x^2+x, x+1} Using Wronskian (5:22) Play Video
21 Annihilator Method 1: Real Linear Factors (14:07) Play Video
22 Example of Annihilator Method: y"-y = sin(2x) (4:37) Play Video
23 Power Series Solution for y"-2y'+y=x, y(0)=0, y'(0)=1 (6:24) Play Video
III. Applications of ODEs: Mass-Spring Systems
24 Mass-Spring Systems 1: Undamped Motion (16:42) Play Video
25 Mass-Spring Systems 2: Underdamped Motion (14:38) Play Video
26 Mass-Spring Systems 3: Critically Damped Motion (9:28) Play Video
IV. The Laplace Transform
27 Laplace Transform of f(t) = 2t-1 (10:42) Play Video
28 Laplace Transform of f(t) = sin(2t) (8:35) Play Video
29 Laplace Transform of f(t) = t sin(2t) (8:34) Play Video
30 Laplace Transform of f(t) = e^{3t}cos(4t) (7:31) Play Video
31 Laplace Transform of f(t) = t^2 e^{2t} cos(3t) (9:53) Play Video
32 Inverse Laplace Transform of (s-1)/s^2(s^2+4) (5:36) Play Video
33 Laplace Transform Solution of y'-3y=e^{2t}, y(0)=2 (5:09) Play Video
34 Laplace Transform Solution of y"-2y'-3y=e^t, y(0) = 0, y'(0) = 1 (10:21) Play Video
35 Laplace Transform of f(t) = 2 on the Interval (1,2) (9:37) Play Video
36 Second Shift Formula for a Piecewise-defined Function (8:10) Play Video
37 Laplace Transform Solution of y'-y=f(t) (Piecewise-Defined) (9:24) Play Video
V. The Convolution Theorem & Fourier Series
38 Example of Convolution Theorem: f(t)=t, g(t)=sin(t) (4:15) Play Video
39 Convolution Theorem for y'-2y=e^t, y(0)=0 (4:24) Play Video
40 Fourier Series: Example of Orthonormal Set of Functions (21:19) Play Video
41 Fourier Series: Example of Parseval's Identity (10:48) Play Video