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Professor Murray
Complex Roots of Second Order Equations
Slide Duration:Table of Contents
I. FirstOrder Equations
Linear Equations
1h 7m 21s
 Intro0:00
 Lesson Objectives0:19
 How to Solve Linear Equations2:54
 Calculate the Integrating Factor2:58
 Changes the Left Side so We Can Integrate Both Sides3:27
 Solving Linear Equations5:32
 Further Notes6:10
 If P(x) is Negative6:26
 Leave Off the Constant9:38
 The C Is Important When Integrating Both Sides of the Equation9:55
 Example 110:29
 Example 222:56
 Example 336:12
 Example 439:24
 Example 544:10
 Example 656:42
Separable Equations
35m 11s
 Intro0:00
 Lesson Objectives0:19
 Some Equations Are Both Linear and Separable So You Can Use Either Technique to Solve Them1:33
 Important to Add C When You Do the Integration2:27
 Example 14:28
 Example 210:45
 Example 314:43
 Example 419:21
 Example 527:23
Slope & Direction Fields
1h 11m 36s
 Intro0:00
 Lesson Objectives0:20
 If You Can Manipulate a Differential Equation Into a Certain Form, You Can Draw a Slope Field Also Known as a Direction Field0:23
 How You Do This0:45
 Solution Trajectories2:49
 Never Cross Each Other3:44
 General Solution to the Differential Equation4:03
 Use an Initial Condition to Find Which Solution Trajectory You Want4:59
 Example 16:52
 Example 214:20
 Example 326:36
 Example 434:21
 Example 546:09
 Example 659:51
Applications, Modeling, & Word Problems of FirstOrder Equations
1h 5m 19s
 Intro0:00
 Lesson Overview0:38
 Mixing1:00
 Population2:49
 Finance3:22
 Set Variables4:39
 Write Differential Equation6:29
 Solve It10:54
 Answer Questions11:47
 Example 113:29
 Example 224:53
 Example 332:13
 Example 442:46
 Example 555:05
Autonomous Equations & Phase Plane Analysis
1h 1m 20s
 Intro0:00
 Lesson Overview0:18
 Autonomous Differential Equations Have the Form y' = f(x)0:21
 Phase Plane Analysis0:48
 y' < 02:56
 y' > 03:04
 If we Perturb the Equilibrium Solutions5:51
 Equilibrium Solutions7:44
 Solutions Will Return to Stable Equilibria8:06
 Solutions Will Tend Away From Unstable Equilibria9:32
 Semistable Equilibria10:59
 Example 111:43
 Example 215:50
 Example 328:27
 Example 431:35
 Example 543:03
 Example 649:01
II. SecondOrder Equations
Distinct Roots of Second Order Equations
28m 44s
 Intro0:00
 Lesson Overview0:36
 Linear Means0:50
 SecondOrder1:15
 Homogeneous1:30
 Constant Coefficient1:55
 Solve the Characteristic Equation2:33
 Roots r1 and r23:43
 To Find c1 and c2, Use Initial Conditions4:50
 Example 15:46
 Example 28:20
 Example 316:20
 Example 418:26
 Example 523:52
Complex Roots of Second Order Equations
31m 49s
 Intro0:00
 Lesson Overview0:15
 Sometimes The Characteristic Equation Has Complex Roots1:12
 Example 13:21
 Example 27:42
 Example 315:25
 Example 418:59
 Example 527:52
Repeated Roots & Reduction of Order
43m 2s
 Intro0:00
 Lesson Overview0:23
 If the Characteristic Equation Has a Double Root1:46
 Reduction of Order3:10
 Example 17:23
 Example 29:20
 Example 314:12
 Example 431:49
 Example 533:21
Undetermined Coefficients of Inhomogeneous Equations
50m 1s
 Intro0:00
 Lesson Overview0:11
 Inhomogeneous Equation Means the Right Hand Side is Not 0 Anymore0:21
 First Solve the Homogeneous Equation1:04
 Find a Particular Solution to the Inhomogeneous Equation Using Undetermined Coefficients2:03
 g(t) vs. Guess for ypar2:42
 If Any Term of Your Guess for ypar Looks Like Any Term of yhom5:07
 Example 17:54
 Example 215:25
 Example 323:45
 Example 433:35
 Example 542:57
Inhomogeneous Equations: Variation of Parameters
49m 22s
 Intro0:00
 Lesson Overview0:31
 Inhomogeneous vs. Homogeneous0:47
 First Solve the Homogeneous Equation1:17
 Notice There is No Coefficient in Front of y''1:27
 Find a Particular Solution to the Inhomogeneous Equation Using Variation of Parameters2:32
 How to Solve4:33
 Hint on Solving the System5:23
 Example 17:27
 Example 217:46
 Example 323:14
 Example 431:49
 Example 536:00
III. Series Solutions
Review of Power Series
57m 38s
 Intro0:00
 Lesson Overview0:36
 Taylor Series Expansion0:37
 Maclaurin Series2:36
 Common Maclaurin Series to Remember From Calculus3:35
 Radius of Convergence7:58
 Ratio Test12:05
 Example 115:18
 Example 220:02
 Example 327:32
 Example 439:33
 Example 545:42
Series Solutions Near an Ordinary Point
1h 20m 28s
 Intro0:00
 Lesson Overview0:49
 Guess a Power Series Solution and Calculate Its Derivatives, Example 11:03
 Guess a Power Series Solution and Calculate Its Derivatives, Example 23:14
 Combine the Series5:00
 Match Exponents on x By Shifting Indices5:11
 Match Starting Indices By Pulling Out Initial Terms5:51
 Find a Recurrence Relation on the Coefficients7:09
 Example 17:46
 Example 219:10
 Example 329:57
 Example 441:46
 Example 557:23
 Example 609:12
Euler Equations
24m 42s
 Intro0:00
 Lesson Overview0:11
 Euler Equation0:15
 Real, Distinct Roots2:22
 Real, Repeated Roots2:37
 Complex Roots2:49
 Example 13:51
 Example 26:20
 Example 38:27
 Example 413:04
 Example 515:31
 Example 618:31
Series Solutions
1h 26m 17s
 Intro0:00
 Lesson Overview0:13
 Singular Point1:17
 Definition: Pole of Order n1:58
 Pole Of Order n2:04
 Regular Singular Point3:25
 Solving Around Regular Singular Points7:08
 Indical Equation7:30
 If the Difference Between the Roots is An Integer8:06
 If the Difference Between the Roots is Not An Integer8:29
 Example 18:47
 Example 214:57
 Example 325:40
 Example 447:23
 Example 509:01
IV. Laplace Transform
Laplace Transforms
41m 52s
 Intro0:00
 Lesson Overview0:09
 Laplace Transform of a Function f(t)0:18
 Laplace Transform is Linear1:04
 Example 11:43
 Example 218:30
 Example 322:06
 Example 428:27
 Example 533:54
Inverse Laplace Transforms
47m 5s
 Intro0:00
 Lesson Overview0:09
 Laplace Transform L{f}0:13
 Run Partial Fractions0:24
 Common Laplace Transforms1:20
 Example 13:24
 Example 29:55
 Example 314:49
 Example 422:03
 Example 533:51
Laplace Transform Initial Value Problems
45m 15s
 Intro0:00
 Lesson Overview0:12
 Start With Initial Value Problem0:14
 Take the Laplace Transform of Both Sides of the Differential Equation0:37
 Plug in the Identities1:20
 Take the Inverse Laplace Transform to Find y2:40
 Example 14:15
 Example 211:30
 Example 317:59
 Example 424:51
 Example 536:05
V. Review of Linear Algebra
Review of Linear Algebra
57m 30s
 Intro0:00
 Lesson Overview0:41
 Matrix0:54
 Determinants4:45
 3x3 Determinants5:08
 Eigenvalues and Eigenvectors7:01
 Eigenvector7:48
 Eigenvalue7:54
 Lesson Overview8:17
 Characteristic Polynomial8:47
 Find Corresponding Eigenvector9:03
 Example 110:19
 Example 216:49
 Example 320:52
 Example 425:34
 Example 535:05
VI. Systems of Equations
Distinct Real Eigenvalues
59m 26s
 Intro0:00
 Lesson Overview1:11
 How to Solve Systems2:48
 Find the Eigenvalues and Their Corresponding Eigenvectors2:50
 General Solution4:30
 Use Initial Conditions to Find c1 and c24:57
 Graphing the Solutions5:20
 Solution Trajectories Tend Towards 0 or ∞ Depending on Whether r1 or r2 are Positive or Negative6:35
 Solution Trajectories Tend Towards the Axis Spanned by the Eigenvector Corresponding to the Larger Eigenvalue7:27
 Example 19:05
 Example 221:06
 Example 326:38
 Example 436:40
 Example 543:26
 Example 651:33
Complex Eigenvalues
1h 3m 54s
 Intro0:00
 Lesson Overview0:47
 Recall That to Solve the System of Linear Differential Equations, We find the Eigenvalues and Eigenvectors0:52
 If the Eigenvalues are Complex, Then They Will Occur in Conjugate Pairs1:13
 Expanding Complex Solutions2:55
 Euler's Formula2:56
 Multiply This Into the Eigenvector, and Separate Into Real and Imaginary Parts1:18
 Graphing Solutions From Complex Eigenvalues5:34
 Example 19:03
 Example 220:48
 Example 328:34
 Example 441:28
 Example 551:21
Repeated Eigenvalues
45m 17s
 Intro0:00
 Lesson Overview0:44
 If the Characteristic Equation Has a Repeated Root, Then We First Find the Corresponding Eigenvector1:14
 Find the Generalized Eigenvector1:25
 Solutions from Repeated Eigenvalues2:22
 Form the Two Principal Solutions and the Two General Solution2:23
 Use Initial Conditions to Solve for c1 and c23:41
 Graphing the Solutions3:53
 Example 18:10
 Example 216:24
 Example 323:25
 Example 431:04
 Example 538:17
VII. Inhomogeneous Systems
Undetermined Coefficients for Inhomogeneous Systems
43m 37s
 Intro0:00
 Lesson Overview0:35
 First Solve the Corresponding Homogeneous System x'=Ax0:37
 Solving the Inhomogeneous System2:32
 Look for a Single Particular Solution xpar to the Inhomogeneous System2:36
 Plug the Guess Into the System and Solve for the Coefficients3:27
 Add the Homogeneous Solution and the Particular Solution to Get the General Solution3:52
 Example 14:49
 Example 29:30
 Example 315:54
 Example 420:39
 Example 529:43
 Example 637:41
Variation of Parameters for Inhomogeneous Systems
1h 8m 12s
 Intro0:00
 Lesson Overview0:37
 Find Two Solutions to the Homogeneous System2:04
 Look for a Single Particular Solution xpar to the inhomogeneous system as follows2:59
 Solutions by Variation of Parameters3:35
 General Solution and Matrix Inversion6:35
 General Solution6:41
 Hint for Finding Ψ16:58
 Example 18:13
 Example 216:23
 Example 332:23
 Example 437:34
 Example 549:00
VIII. Numerical Techniques
Euler's Method
45m 30s
 Intro0:00
 Lesson Overview0:32
 Euler's Method is a Way to Find Numerical Approximations for Initial Value Problems That We Cannot Solve Analytically0:34
 Based on Drawing Lines Along Slopes in a Direction Field1:18
 Formulas for Euler's Method1:57
 Example 14:47
 Example 214:45
 Example 324:03
 Example 433:01
 Example 537:55
RungeKutta & The Improved Euler Method
41m 4s
 Intro0:00
 Lesson Overview0:43
 RungeKutta is Know as the Improved Euler Method0:46
 More Sophisticated Than Euler's Method1:09
 It is the Fundamental Algorithm Used in Most Professional Software to Solve Differential Equations1:16
 Order 2 RungeKutta Algorithm1:45
 RungeKutta Order 2 Algorithm2:09
 Example 14:57
 Example 210:57
 Example 319:45
 Example 424:35
 Example 531:39
IX. Partial Differential Equations
Review of Partial Derivatives
38m 22s
 Intro0:00
 Lesson Overview1:04
 Partial Derivative of u with respect to x1:37
 Geometrically, ux Represents the Slope As You Walk in the xdirection on the Surface2:47
 Computing Partial Derivatives3:46
 Algebraically, to Find ux You Treat The Other Variable t as a Constant and Take the Derivative with Respect to x3:49
 Second Partial Derivatives4:16
 Clairaut's Theorem Says that the Two 'Mixed Partials' Are Always Equal5:21
 Example 15:34
 Example 27:40
 Example 311:17
 Example 414:23
 Example 531:55
The Heat Equation
44m 40s
 Intro0:00
 Lesson Overview0:28
 Partial Differential Equation0:33
 Most Common Ones1:17
 Boundary Value Problem1:41
 Common Partial Differential Equations3:41
 Heat Equation4:04
 Wave Equation5:44
 Laplace's Equation7:50
 Example 18:35
 Example 214:21
 Example 321:04
 Example 425:54
 Example 535:12
Separation of Variables
57m 44s
 Intro0:00
 Lesson Overview0:26
 Separation of Variables is a Technique for Solving Some Partial Differential Equations0:29
 Separation of Variables2:35
 Try to Separate the Variables2:38
 If You Can, Then Both Sides Must Be Constant2:52
 Reorganize These Intro Two Ordinary Differential Equations3:05
 Example 14:41
 Example 211:06
 Example 318:30
 Example 425:49
 Example 532:53
Fourier Series
1h 24m 33s
 Intro0:00
 Lesson Overview0:38
 Fourier Series0:42
 Find the Fourier Coefficients by the Formulas2:05
 Notes on Fourier Series3:34
 Formula Simplifies3:35
 Function Must be Periodic4:23
 Even and Odd Functions5:37
 Definition5:45
 Examples6:03
 Even and Odd Functions and Fourier Series9:47
 If f is Even9:52
 If f is Odd11:29
 Extending Functions12:46
 If We Want a Cosine Series14:13
 If We Wants a Sine Series15:20
 Example 117:39
 Example 243:23
 Example 351:14
 Example 401:52
 Example 511:53
Solution of the Heat Equation
47m 41s
 Intro0:00
 Lesson Overview0:22
 Solving the Heat Equation1:03
 Procedure for the Heat Equation3:29
 Extend So That its Fourier Series Will Have Only Sines3:57
 Find the Fourier Series for f(x)4:19
 Example 15:21
 Example 28:08
 Example 317:42
 Example 425:13
 Example 528:53
 Example 642:22
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For more information, please see full course syllabus of Differential Equations
For more information, please see full course syllabus of Differential Equations
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1 answer
Last reply by: Dr. William Murray
Thu Dec 22, 2016 11:22 AM
Post by Marwa Alzubaidi on December 20, 2016
I have problem I want solve by matlab Could you helpe me?
1 answer
Last reply by: Dr. William Murray
Mon Mar 30, 2015 1:05 PM
Post by Ulysses Hickey on March 29, 2015
Where does the "i" in the imaginary root end up going ?
 In the process demonstrated it seems as if the i is dropped completely.
But further study in my own text indicates that the i is essential to the solution.
Please Explain.
1 answer
Last reply by: Dr. William Murray
Fri Feb 6, 2015 12:28 PM
Post by Salah Issa on February 4, 2015
Thank you Dr.Murray,
I am really thankful
Regards,
Salah Issa
3 answers
Last reply by: Dr. William Murray
Fri Feb 6, 2015 12:29 PM
Post by Salah Issa on January 17, 2015
Dear Dr.Muray,
Kindly, can you explain what is the reason behind sine and cosine in the general solution?