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Differential Equations Online Course Dr. William Murray
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Level Advanced
30 Lessons (26hr : 33min)
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4.8 632 Ratings & 38 Reviews
Audio: English
English

Join Dr. William Murray in his Differential Equations online course complete with clear explanations of theory and a wide array of helpful insights. Each lesson also includes several step-by-step practice problems like the ones you will see on homework and tests.

Table of Contents
Course Details
About Dr. Murray

Section 1: First-Order 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 First-Order 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: Second-Order Equations
	Distinct Roots of Second Order Equations			28:44
		Intro		0:00
		Lesson Overview		0:36
			Linear Means	0:50
			Second-Order	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 Ψ-1	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
	Runge-Kutta & The Improved Euler Method			41:04
		Intro		0:00
		Lesson Overview		0:43
			Runge-Kutta 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 Runge-Kutta Algorithm	1:45
		Runge-Kutta 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 x-direction 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

Duration: 26 hours, 33 minutes

Number of Lessons: 30

This course is perfect for the college student taking Differential Equations and will help you understand & solve problems from all over biology, physics, chemistry, and engineering. No more getting stuck in one of the hardest college math courses.

Additional Features:

Free Sample Lessons
Closed Captioning (CC)
Practice Questions
Downloadable Lecture Slides
Study Guides
Instructor Comments

Topics Include:

Separable Equations
Direction Fields
Second Order Equations
Euler Equations
Laplace Transforms
Eigenvalues & Eigenvectors
Inhomogenous Systems
Partial Derivatives
Heat Equation
Fourier Series

Dr. Murray received his Ph.D from UC Berkeley, his BS from Georgetown University, and has been teaching in the university setting for 15+ years.

Student Testimonials:

"I'm very glad there are still great professors out there such as yourself who know exactly how to teach. I'm very grateful." — Mateusz M.

“Hi Dr. Murray, your lectures are great and clear.” — Kyung Yeop K.

“Excellent! Wunderbar!” — William D.

“Great teaching professor. I have learned so much from your videos. I cannot thank you enough.” — Eddy N.

“Let me first say, thanks so much for making these as quickly as you did, I really didn't expect they'd be complete until after my DE course. These are fantastic, I'm very grateful.” — Caleb L.

Visit Dr. Murray’s page

Student Feedback

4.8

38 Reviews

50%

By Manuel Gonzalez parra•August 11, 2017

Can Laplace transforms be used on differential equations not in the form given at the beginning of the video? Because for those equations I’d rather use the method of undetermined coefficients covered earlier. Thank you! Great videos!

By Silvia Gonzalez•June 28, 2016

Thank you.

By Silvia Gonzalez•June 28, 2016

Thank you! Yes, it made lots of sense.

By David LÃ¶fqvist•March 6, 2016

Hi! Great lecture, and yes I would also love to see more topics, but that's not my reason for commenting. I'm wondering if you could give some physical example of a semistable equalibrium?

By Jonathan Snow•January 1, 2016

Oh, I see it now, thank you, that makes sense.