Math 252: Applied Differential Equations

Lecture Notes

Chapter 1. Introduction to Differential Equations

• We worked through the Section 1.1 Definitions Handout which you can find in the Handouts below. Also, Section 1.1: Definitions and Terminology
• Section 1.2: Initial-Value Problems

Chapter 2. First-Order Differential Equations

• Section 2.2: Separable Variables
• Section 2.3: Linear Equations. Also, see Handouts below for the Explainer on how the integrating factor works.
• Section 2.4: Exact Equations. Also, see Handouts below for the Explainer on how the integrating factor works.
• Sections 2.2 to 2.4: Methods Covered Thus Far
• Section 2.5: Solutions by Substitution. Also, see Handouts below for the Homogeneous Example using both substitutions.
• See also the handouts below for Steps to Determine Type of DE and the Name that DE handouts.

• Test Dates Announcement

Chapter 3. Modeling with First-Order Differential Equations

• Section 3.1: Linear Models

• Test 1 covers the material up to this point.

Chapter 4. Higher-Order Differential Equations

• Section 4.1: Preliminary Theory - Linear Equations. Also, see Handouts below for the Preliminary Theory handout.
• Section 4.2: Reduction of Order. Also, see Handouts below for the Explainer on how finding the second solution works with Reduction of Order.
• Section 4.3: Homogeneous Linear Equations with Constant Coefficients. Also, see Handouts below for
  • Explainer on the Methodology
  • Explainer on Complex Solutions
  • Susie's Handout on Higher Order Homogeneous DEs
  • Graphs of Solutions
• Section 4.4: Undetermined Coefficients

• Review for Test 1

• Section 4.6: Variation of Parameters
• Section 4.7: Cauchy-Euler Equations

Chapter 5. Modeling with Higher-Order Differential Equations

• Section 5.1: Linear Models: Initial-Value Problems

• Test 2 covers the material up to this point.

Chapter 6. Series Solutions of Linear Equations

• Section 6.1: Review of Power Series
• Section 6.2: Solutions about Ordinary Points

• Review for Test 2

Chapter 7. Laplace Transforms

• Section 7.1: Definition of the Laplace Transform
• Section 7.2: Inverse Transforms and Transforms of Derivatives
• Section 7.3: Operational Properties I

• Test 3 covers the material up to this point.

• Section 7.4: Operational Properties II – We are only doing the derivatives of transforms. Omit convolutions.
• Section 7.5: The Dirac Delta Function

Chapter 8. Systems of Linear First-Order Differential Equations

• Section 8.1: Preliminary Theory - Linear Systems
• Section 8.2: Homogeneous Linear Systems

Videos

Section 7.2:

• Finishing up Inverse Laplace example from last in-person lecture (6:44 min)
• Laplace Transforms of Derivatives (7:52 min)
• Solving IVPs using Laplace: Example 1 (4:14 min)
• Solving IVPs using Laplace: Example 2 (10:26 min)
• Solving IVPs using Laplace: Example 3 (10:31 min)

Section 7.3:

• Laplace transforms of translations on the s-axis (6:08 min)
• Examples of Laplace transforms of translations on the s-axis (4:01 min)
• Examples of inverse Laplace transforms of translations on the s-axis (6:08 min)
• More examples of inverse Laplace transforms of translations on the s-axis (11:19 min)
• Solving IVPs using Laplace and translations on s-axis: Example 1 (7:10 min)
• Solving IVPs using Laplace and translations on s-axis: Example 2 (8:39 min)

Section 7.4:

• Derivatives of Laplace transforms (10:03 min)
• Solving IVPs using derivatives of Laplace transforms (7:15 min)

Section 8.1:

• Intro to systems of linear first-order DEs (7:27 min)
• Verifying solutions to systems of linear first-order DEs (7:50 min)

Section 8.2:

• Homogeneous systems of linear first-order DEs - theory (8:36 min)
• Homogeneous systems of linear first-order DEs - distinct eigenvalues, 2x2 example (11:15 min)
• Homogeneous systems of linear first-order DEs - repeated eigenvalues, 3x3 example (8:32 min)
• Homogeneous systems of linear first-order DEs - complex eigenvalues, theory (11:49 min)
• Homogeneous systems of linear first-order DEs - complex eigenvalues, 2x2 example (8:24 min)

Handouts, In-class Visuals, and Extras

Chapter 1:

• Section 1.1: Definitions

Chapter 2:

• Section 2.3: Explainer on I.F. for 1st order linear DEs
• Section 2.3: Explainer on I.F. for exact DEs
• Section 2.3: Full example of Homogeneous using Multiple Methods
• Chapter 2: Steps to Determine Type of DE
• Chapter 2: Name that DE!

Chapter 4:

• Section 4.1: Preliminary Theory on Higher Order Linear DEs plus the annotated version
• Section 4.2: Explainer on Reduction of Order
• Section 4.3:
  • Explainer on Methodology
  • Explainer on Complex Solutions
  • Susie's Handout on Higher Order Homogeneous DEs
  • Graphs of Solutions and Annotated Version
• Section 4.4:
  • Gilles' handout on the Method of Undetermined Coefficients
  • Graphs of Solutions and Annotated Version
  • Beat Frequency Demo
• Section 4.6:
  • Explainer on Methodology
  • Explainer on Why No +C in u's
• Section 4.7:
  • Gilles' handout on the Constant coefficients and Cauchy-Euler equations
  • Explainer on Cauchy-Euler equations
  • Explainer on Quick Constant Coeffs vs. Cauchy-Euler

Chapter 5:

• Section 5.1: Gilles' handout on Linear combination of Sine and Cosine
• Section 5.1: A student asked me what it would look like if you had damped oscillation plus an external force which is sinusoidal but with a slightly different frequency. As we've seen, damped oscillation looks like this. Undamped oscillation plus an external sinusoidal force can give a beat frequency. Damped oscillation with beats can look something like this (depends on the constants), but as you can see there is an initial "beat" state, which decays away until only the steady state solution (from the external force) is left.

Chapter 6:

• Section 6.1: Quick Review of Power Series

Chapter 7:

• Gilles' table of Laplace transforms
• Gilles' problem book on Laplace transforms

Chapter 8:

• Summary of solutions for homogeneous linear systems of DEs.

Review Materials

• Useful Properties of Logs
• Completing the Square – if you need a reminder on how to complete the square, then this is the worksheet for you!
• Gilles' Review of Eigenvalues and Eigenvectors