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Variation of Parameters for Inhomogeneous Systems

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Section 1: First-Order 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 First-Order 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

Section 2: Second-Order Equations

Distinct Roots of Second Order Equations

28m 44s

- Intro0:00
- Lesson Overview0:36
- Linear Means0:50
- Second-Order1:15
- 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

Section 3: 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 61:09:12

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 51:09:01

Section 4: 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

Section 5: 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

Section 6: 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

Section 7: 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 Ψ-16:58
- Example 18:13
- Example 216:23
- Example 332:23
- Example 437:34
- Example 549:00

Section 8: 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

Runge-Kutta & The Improved Euler Method

41m 4s

- Intro0:00
- Lesson Overview0:43
- Runge-Kutta 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 Runge-Kutta Algorithm1:45
- Runge-Kutta Order 2 Algorithm2:09
- Example 14:57
- Example 210:57
- Example 319:45
- Example 424:35
- Example 531:39

Section 9: 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 x-direction 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 41:01:52
- Example 51:11:53

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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5 answers

Last reply by: Dr. William Murray

Wed Mar 11, 2015 3:02 PM

Post by Jennie Hill on February 19, 2015

Hi,

Question for you, at ~55:40, when you integrate -2sin(t)cos(t), why did you choose to use the "w"-substitution as opposed to simplifying first with the Double Angle Property?

My students and I would appreciate an explanation. I'm assuming it's to keep the angles the same.

Thanks!

1 answer

Last reply by: Dr. William Murray

Fri May 30, 2014 3:48 PM

Post by Astrit Imeri on May 29, 2014

The first one

1 answer

Last reply by: Dr. William Murray

Fri May 30, 2014 3:48 PM

Post by Astrit Imeri on May 29, 2014

Checking past papers, this question was on the final exam of the last semester in the university that I am attending.