Section 1: FirstOrder Equations 

Linear Equations 
1:07:21 
 
Intro 
0:00  
 
Lesson Objectives 
0:19  
 
How to Solve Linear Equations 
2:54  
 
 Calculate the Integrating Factor 
2:58  
 
 Changes the Left Side so We Can Integrate Both Sides 
3:27  
 
 Solving Linear Equations 
5:32  
 
Further Notes 
6:10  
 
 If P(x) is Negative 
6:26  
 
 Leave Off the Constant 
9:38  
 
 The C Is Important When Integrating Both Sides of the Equation 
9:55  
 
Example 1 
10:29  
 
Example 2 
22:56  
 
Example 3 
36:12  
 
Example 4 
39:24  
 
Example 5 
44:10  
 
Example 6 
56:42  

Separable Equations 
35:11 
 
Intro 
0:00  
 
Lesson Objectives 
0:19  
 
 Some Equations Are Both Linear and Separable So You Can Use Either Technique to Solve Them 
1:33  
 
 Important to Add C When You Do the Integration 
2:27  
 
Example 1 
4:28  
 
Example 2 
10:45  
 
Example 3 
14:43  
 
Example 4 
19:21  
 
Example 5 
27:23  

Slope & Direction Fields 
1:11:36 
 
Intro 
0:00  
 
Lesson Objectives 
0:20  
 
 If You Can Manipulate a Differential Equation Into a Certain Form, You Can Draw a Slope Field Also Known as a Direction Field 
0:23  
 
 How You Do This 
0:45  
 
Solution Trajectories 
2:49  
 
 Never Cross Each Other 
3:44  
 
 General Solution to the Differential Equation 
4:03  
 
 Use an Initial Condition to Find Which Solution Trajectory You Want 
4:59  
 
Example 1 
6:52  
 
Example 2 
14:20  
 
Example 3 
26:36  
 
Example 4 
34:21  
 
Example 5 
46:09  
 
Example 6 
59:51  

Applications, Modeling, & Word Problems of FirstOrder Equations 
1:05:19 
 
Intro 
0:00  
 
Lesson Overview 
0:38  
 
 Mixing 
1:00  
 
 Population 
2:49  
 
 Finance 
3:22  
 
 Set Variables 
4:39  
 
 Write Differential Equation 
6:29  
 
 Solve It 
10:54  
 
 Answer Questions 
11:47  
 
Example 1 
13:29  
 
Example 2 
24:53  
 
Example 3 
32:13  
 
Example 4 
42:46  
 
Example 5 
55:05  

Autonomous Equations & Phase Plane Analysis 
1:01:20 
 
Intro 
0:00  
 
Lesson Overview 
0:18  
 
 Autonomous Differential Equations Have the Form y' = f(x) 
0:21  
 
 Phase Plane Analysis 
0:48  
 
 y' < 0 
2:56  
 
 y' > 0 
3:04  
 
 If we Perturb the Equilibrium Solutions 
5:51  
 
 Equilibrium Solutions 
7:44  
 
 Solutions Will Return to Stable Equilibria 
8:06  
 
 Solutions Will Tend Away From Unstable Equilibria 
9:32  
 
 Semistable Equilibria 
10:59  
 
Example 1 
11:43  
 
Example 2 
15:50  
 
Example 3 
28:27  
 
Example 4 
31:35  
 
Example 5 
43:03  
 
Example 6 
49:01  
Section 2: SecondOrder Equations 

Distinct Roots of Second Order Equations 
28:44 
 
Intro 
0:00  
 
Lesson Overview 
0:36  
 
 Linear Means 
0:50  
 
 SecondOrder 
1:15  
 
 Homogeneous 
1:30  
 
 Constant Coefficient 
1:55  
 
 Solve the Characteristic Equation 
2:33  
 
 Roots r1 and r2 
3:43  
 
 To Find c1 and c2, Use Initial Conditions 
4:50  
 
Example 1 
5:46  
 
Example 2 
8:20  
 
Example 3 
16:20  
 
Example 4 
18:26  
 
Example 5 
23:52  

Complex Roots of Second Order Equations 
31:49 
 
Intro 
0:00  
 
Lesson Overview 
0:15  
 
 Sometimes The Characteristic Equation Has Complex Roots 
1:12  
 
Example 1 
3:21  
 
Example 2 
7:42  
 
Example 3 
15:25  
 
Example 4 
18:59  
 
Example 5 
27:52  

Repeated Roots & Reduction of Order 
43:02 
 
Intro 
0:00  
 
Lesson Overview 
0:23  
 
 If the Characteristic Equation Has a Double Root 
1:46  
 
 Reduction of Order 
3:10  
 
Example 1 
7:23  
 
Example 2 
9:20  
 
Example 3 
14:12  
 
Example 4 
31:49  
 
Example 5 
33:21  

Undetermined Coefficients of Inhomogeneous Equations 
50:01 
 
Intro 
0:00  
 
Lesson Overview 
0:11  
 
 Inhomogeneous Equation Means the Right Hand Side is Not 0 Anymore 
0:21  
 
 First Solve the Homogeneous Equation 
1:04  
 
 Find a Particular Solution to the Inhomogeneous Equation Using Undetermined Coefficients 
2:03  
 
 g(t) vs. Guess for ypar 
2:42  
 
 If Any Term of Your Guess for ypar Looks Like Any Term of yhom 
5:07  
 
Example 1 
7:54  
 
Example 2 
15:25  
 
Example 3 
23:45  
 
Example 4 
33:35  
 
Example 5 
42:57  

Inhomogeneous Equations: Variation of Parameters 
49:22 
 
Intro 
0:00  
 
Lesson Overview 
0:31  
 
 Inhomogeneous vs. Homogeneous 
0:47  
 
 First Solve the Homogeneous Equation 
1:17  
 
 Notice There is No Coefficient in Front of y'' 
1:27  
 
 Find a Particular Solution to the Inhomogeneous Equation Using Variation of Parameters 
2:32  
 
 How to Solve 
4:33  
 
 Hint on Solving the System 
5:23  
 
Example 1 
7:27  
 
Example 2 
17:46  
 
Example 3 
23:14  
 
Example 4 
31:49  
 
Example 5 
36:00  
Section 3: Series Solutions 

Review of Power Series 
57:38 
 
Intro 
0:00  
 
Lesson Overview 
0:36  
 
 Taylor Series Expansion 
0:37  
 
 Maclaurin Series 
2:36  
 
 Common Maclaurin Series to Remember From Calculus 
3:35  
 
 Radius of Convergence 
7:58  
 
 Ratio Test 
12:05  
 
Example 1 
15:18  
 
Example 2 
20:02  
 
Example 3 
27:32  
 
Example 4 
39:33  
 
Example 5 
45:42  

Series Solutions Near an Ordinary Point 
1:20:28 
 
Intro 
0:00  
 
Lesson Overview 
0:49  
 
 Guess a Power Series Solution and Calculate Its Derivatives, Example 1 
1:03  
 
 Guess a Power Series Solution and Calculate Its Derivatives, Example 2 
3:14  
 
 Combine the Series 
5:00  
 
 Match Exponents on x By Shifting Indices 
5:11  
 
 Match Starting Indices By Pulling Out Initial Terms 
5:51  
 
 Find a Recurrence Relation on the Coefficients 
7:09  
 
Example 1 
7:46  
 
Example 2 
19:10  
 
Example 3 
29:57  
 
Example 4 
41:46  
 
Example 5 
57:23  
 
Example 6 
69:12  

Euler Equations 
24:42 
 
Intro 
0:00  
 
Lesson Overview 
0:11  
 
 Euler Equation 
0:15  
 
 Real, Distinct Roots 
2:22  
 
 Real, Repeated Roots 
2:37  
 
 Complex Roots 
2:49  
 
Example 1 
3:51  
 
Example 2 
6:20  
 
Example 3 
8:27  
 
Example 4 
13:04  
 
Example 5 
15:31  
 
Example 6 
18:31  

Series Solutions 
1:26:17 
 
Intro 
0:00  
 
Lesson Overview 
0:13  
 
 Singular Point 
1:17  
 
Definition: Pole of Order n 
1:58  
 
 Pole Of Order n 
2:04  
 
 Regular Singular Point 
3:25  
 
Solving Around Regular Singular Points 
7:08  
 
 Indical Equation 
7:30  
 
 If the Difference Between the Roots is An Integer 
8:06  
 
 If the Difference Between the Roots is Not An Integer 
8:29  
 
Example 1 
8:47  
 
Example 2 
14:57  
 
Example 3 
25:40  
 
Example 4 
47:23  
 
Example 5 
69:01  
Section 4: Laplace Transform 

Laplace Transforms 
41:52 
 
Intro 
0:00  
 
Lesson Overview 
0:09  
 
 Laplace Transform of a Function f(t) 
0:18  
 
 Laplace Transform is Linear 
1:04  
 
Example 1 
1:43  
 
Example 2 
18:30  
 
Example 3 
22:06  
 
Example 4 
28:27  
 
Example 5 
33:54  

Inverse Laplace Transforms 
47:05 
 
Intro 
0:00  
 
Lesson Overview 
0:09  
 
 Laplace Transform L{f} 
0:13  
 
 Run Partial Fractions 
0:24  
 
Common Laplace Transforms 
1:20  
 
Example 1 
3:24  
 
Example 2 
9:55  
 
Example 3 
14:49  
 
Example 4 
22:03  
 
Example 5 
33:51  

Laplace Transform Initial Value Problems 
45:15 
 
Intro 
0:00  
 
Lesson Overview 
0:12  
 
 Start With Initial Value Problem 
0:14  
 
 Take the Laplace Transform of Both Sides of the Differential Equation 
0:37  
 
 Plug in the Identities 
1:20  
 
 Take the Inverse Laplace Transform to Find y 
2:40  
 
 Example 1 
4:15  
 
 Example 2 
11:30  
 
 Example 3 
17:59  
 
 Example 4 
24:51  
 
 Example 5 
36:05  
Section 5: Review of Linear Algebra 

Review of Linear Algebra 
57:30 
 
Intro 
0:00  
 
Lesson Overview 
0:41  
 
 Matrix 
0:54  
 
 Determinants 
4:45  
 
3x3 Determinants 
5:08  
 
Eigenvalues and Eigenvectors 
7:01  
 
 Eigenvector 
7:48  
 
 Eigenvalue 
7:54  
 
Lesson Overview 
8:17  
 
 Characteristic Polynomial 
8:47  
 
 Find Corresponding Eigenvector 
9:03  
 
Example 1 
10:19  
 
Example 2 
16:49  
 
Example 3 
20:52  
 
Example 4 
25:34  
 
Example 5 
35:05  
Section 6: Systems of Equations 

Distinct Real Eigenvalues 
59:26 
 
Intro 
0:00  
 
Lesson Overview 
1:11  
 
How to Solve Systems 
2:48  
 
 Find the Eigenvalues and Their Corresponding Eigenvectors 
2:50  
 
 General Solution 
4:30  
 
 Use Initial Conditions to Find c1 and c2 
4:57  
 
Graphing the Solutions 
5:20  
 
 Solution Trajectories Tend Towards 0 or ∞ Depending on Whether r1 or r2 are Positive or Negative 
6:35  
 
 Solution Trajectories Tend Towards the Axis Spanned by the Eigenvector Corresponding to the Larger Eigenvalue 
7:27  
 
Example 1 
9:05  
 
Example 2 
21:06  
 
Example 3 
26:38  
 
Example 4 
36:40  
 
Example 5 
43:26  
 
Example 6 
51:33  

Complex Eigenvalues 
1:03:54 
 
Intro 
0:00  
 
Lesson Overview 
0:47  
 
 Recall That to Solve the System of Linear Differential Equations, We find the Eigenvalues and Eigenvectors 
0:52  
 
 If the Eigenvalues are Complex, Then They Will Occur in Conjugate Pairs 
1:13  
 
Expanding Complex Solutions 
2:55  
 
 Euler's Formula 
2:56  
 
 Multiply This Into the Eigenvector, and Separate Into Real and Imaginary Parts 
1:18  
 
Graphing Solutions From Complex Eigenvalues 
5:34  
 
Example 1 
9:03  
 
Example 2 
20:48  
 
Example 3 
28:34  
 
Example 4 
41:28  
 
Example 5 
51:21  

Repeated Eigenvalues 
45:17 
 
Intro 
0:00  
 
Lesson Overview 
0:44  
 
 If the Characteristic Equation Has a Repeated Root, Then We First Find the Corresponding Eigenvector 
1:14  
 
 Find the Generalized Eigenvector 
1:25  
 
Solutions from Repeated Eigenvalues 
2:22  
 
 Form the Two Principal Solutions and the Two General Solution 
2:23  
 
 Use Initial Conditions to Solve for c1 and c2 
3:41  
 
Graphing the Solutions 
3:53  
 
Example 1 
8:10  
 
Example 2 
16:24  
 
Example 3 
23:25  
 
Example 4 
31:04  
 
Example 5 
38:17  
Section 7: Inhomogeneous Systems 

Undetermined Coefficients for Inhomogeneous Systems 
43:37 
 
Intro 
0:00  
 
Lesson Overview 
0:35  
 
 First Solve the Corresponding Homogeneous System x'=Ax 
0:37  
 
Solving the Inhomogeneous System 
2:32  
 
 Look for a Single Particular Solution xpar to the Inhomogeneous System 
2:36  
 
 Plug the Guess Into the System and Solve for the Coefficients 
3:27  
 
 Add the Homogeneous Solution and the Particular Solution to Get the General Solution 
3:52  
 
Example 1 
4:49  
 
Example 2 
9:30  
 
Example 3 
15:54  
 
Example 4 
20:39  
 
Example 5 
29:43  
 
Example 6 
37:41  

Variation of Parameters for Inhomogeneous Systems 
1:08:12 
 
Intro 
0:00  
 
Lesson Overview 
0:37  
 
 Find Two Solutions to the Homogeneous System 
2:04  
 
 Look for a Single Particular Solution xpar to the inhomogeneous system as follows 
2:59  
 
Solutions by Variation of Parameters 
3:35  
 
General Solution and Matrix Inversion 
6:35  
 
 General Solution 
6:41  
 
 Hint for Finding Ψ1 
6:58  
 
Example 1 
8:13  
 
Example 2 
16:23  
 
Example 3 
32:23  
 
Example 4 
37:34  
 
Example 5 
49:00  
Section 8: Numerical Techniques 

Euler's Method 
45:30 
 
Intro 
0:00  
 
Lesson Overview 
0:32  
 
 Euler's Method is a Way to Find Numerical Approximations for Initial Value Problems That We Cannot Solve Analytically 
0:34  
 
 Based on Drawing Lines Along Slopes in a Direction Field 
1:18  
 
Formulas for Euler's Method 
1:57  
 
Example 1 
4:47  
 
Example 2 
14:45  
 
Example 3 
24:03  
 
Example 4 
33:01  
 
Example 5 
37:55  

RungeKutta & The Improved Euler Method 
41:04 
 
Intro 
0:00  
 
Lesson Overview 
0:43  
 
 RungeKutta is Know as the Improved Euler Method 
0:46  
 
 More Sophisticated Than Euler's Method 
1:09  
 
 It is the Fundamental Algorithm Used in Most Professional Software to Solve Differential Equations 
1:16  
 
 Order 2 RungeKutta Algorithm 
1:45  
 
RungeKutta Order 2 Algorithm 
2:09  
 
Example 1 
4:57  
 
Example 2 
10:57  
 
Example 3 
19:45  
 
Example 4 
24:35  
 
Example 5 
31:39  
Section 9: Partial Differential Equations 

Review of Partial Derivatives 
38:22 
 
Intro 
0:00  
 
Lesson Overview 
1:04  
 
 Partial Derivative of u with respect to x 
1:37  
 
 Geometrically, ux Represents the Slope As You Walk in the xdirection on the Surface 
2:47  
 
Computing Partial Derivatives 
3:46  
 
 Algebraically, to Find ux You Treat The Other Variable t as a Constant and Take the Derivative with Respect to x 
3:49  
 
 Second Partial Derivatives 
4:16  
 
 Clairaut's Theorem Says that the Two 'Mixed Partials' Are Always Equal 
5:21  
 
Example 1 
5:34  
 
Example 2 
7:40  
 
Example 3 
11:17  
 
Example 4 
14:23  
 
Example 5 
31:55  

The Heat Equation 
44:40 
 
Intro 
0:00  
 
Lesson Overview 
0:28  
 
 Partial Differential Equation 
0:33  
 
 Most Common Ones 
1:17  
 
 Boundary Value Problem 
1:41  
 
Common Partial Differential Equations 
3:41  
 
 Heat Equation 
4:04  
 
 Wave Equation 
5:44  
 
 Laplace's Equation 
7:50  
 
Example 1 
8:35  
 
Example 2 
14:21  
 
Example 3 
21:04  
 
Example 4 
25:54  
 
Example 5 
35:12  

Separation of Variables 
57:44 
 
Intro 
0:00  
 
Lesson Overview 
0:26  
 
 Separation of Variables is a Technique for Solving Some Partial Differential Equations 
0:29  
 
Separation of Variables 
2:35  
 
 Try to Separate the Variables 
2:38  
 
 If You Can, Then Both Sides Must Be Constant 
2:52  
 
 Reorganize These Intro Two Ordinary Differential Equations 
3:05  
 
Example 1 
4:41  
 
Example 2 
11:06  
 
Example 3 
18:30  
 
Example 4 
25:49  
 
Example 5 
32:53  

Fourier Series 
1:24:33 
 
Intro 
0:00  
 
Lesson Overview 
0:38  
 
 Fourier Series 
0:42  
 
 Find the Fourier Coefficients by the Formulas 
2:05  
 
Notes on Fourier Series 
3:34  
 
 Formula Simplifies 
3:35  
 
 Function Must be Periodic 
4:23  
 
Even and Odd Functions 
5:37  
 
 Definition 
5:45  
 
 Examples 
6:03  
 
Even and Odd Functions and Fourier Series 
9:47  
 
 If f is Even 
9:52  
 
 If f is Odd 
11:29  
 
Extending Functions 
12:46  
 
 If We Want a Cosine Series 
14:13  
 
 If We Wants a Sine Series 
15:20  
 
Example 1 
17:39  
 
Example 2 
43:23  
 
Example 3 
51:14  
 
Example 4 
61:52  
 
Example 5 
71:53  

Solution of the Heat Equation 
47:41 
 
Intro 
0:00  
 
Lesson Overview 
0:22  
 
Solving the Heat Equation 
1:03  
 
Procedure for the Heat Equation 
3:29  
 
 Extend So That its Fourier Series Will Have Only Sines 
3:57  
 
 Find the Fourier Series for f(x) 
4:19  
 
Example 1 
5:21  
 
Example 2 
8:08  
 
Example 3 
17:42  
 
Example 4 
25:13  
 
Example 5 
28:53  
 
Example 6 
42:22  