**Math 4B**

Differential Equations

- Khan Academy: What is a differential equation?
Video - 11:02 Introduction to differential equations and the terms order and linear/nonlinear - Paul's Notes: Definitions
Text Definitions for order, linear, initial value problem, implicit/explicit solutions - Paul's Notes: Direction Fields
Text Introduction to direction fields, isoclines, and determining long term behavior - Direction Field Plotter
Interactive Allows you to enter in function and will plot direction field and solutions - Mathispower4u: Slope Fields
Video - 9:01 Intro to slope fields and examples - CLAS Lee DeAnda's webpage: Slope Fields
Interactive Step by step explanation of slope fields and isoclines - CalculusApplets: Slope Fields
Interactive Interactive applet with example slope fields - MIT: Isoclines
Interactive Interactive applet that plots example slope fields and isoclines - Khan Academy: Separable Differential Equations
Video - 12:00 Examples using separation of variables, one of which has a solution left in implicit form See second video for more examples - Paul's Notes: Separable Equations
Text Description of separation of variables method and some examples - PatrickJMT: Solving Separable First Order Differential Equations
Video - 5:08 Example using the separation of variables method - PatrickJMT: Solving Separable First Order Differential Equations Ex 2
Video - 5:18 Second example using the separation of variables metod - MIT: Basic DE's and Separable Equations
Video / PDF Video, course notes, and practice problems on separation of variables - Paul's Notes: Linear Equations
Text Description of integrating factor method and some examples - PatrickJMT: First Order Linear Differential Equations
Video - 5:48 Example using the integrating factor method - PatrickJMT: First Order Linear Differential Equations Ex 2
Video - 3:30 Second example using the integrating factor method - MIT: First Order Linear ODE's
Video / PDF Introduction to linear ODEs and the superposition principle Can ignore content on systems - MIT: Solution by Integrating Factors
Video / PDF Video lectures, course notes, and practice problems on integrating factor method - CLAS Lee DeAnda's webpage: Diff Eqn
Text Supplementary material to 34B CLAS groups on diff eqn - CalculusApplets: Growth, Decay, and the Logistic Equation
Interactive Gives slope field and sample solutions to growth/decay and logistic equations - Mathispower4u: Introduction to Differential Equations
Video - 8:12 Introduction to differential equations - Mathispower4u: Differential Equations and the exponential function
Video - 8:07 Introduction to growth and decay differential equations - Mathispower4u: Newton's Law of Cooling
Video - 8:25 Introduction to Newton's Law of Cooling and examples - University of British Columbia: Initial Value Problems for Growth and Decay
Text Introduction and examples for growth/decay - University of British Columbia: Newton's Law of Cooling
Text In depth explanation of Newton's Law of Cooling - University of British Columbia: The Logistic Equation
Text Step-by-step proof of the solution to the logistic equation - PatrickJMT: The Logistic Equation, Example 1, part 1
Video - 7:04 Population growth example requiring the logistic equation - Paul's Notes: Modeling with First Order DE's
Text Various applications of first order DE's Most relevant may be the first example on mixing - PatrickJMT: Mixing Problems and Separable Differential Equations
Video - 10:09 Mixing example

Direction Fields

Separable Differential Equations

Linear Differential Equations

Applications

- Paul's Notes: Basic Concepts
Text Introduction to second order DE's and the principle of superposition in this setting - Khan Academy: Linear Homogenous Equations 1
Video - 9:44 Introduction to second order DE's (linear homogenous case) - PatrickJMT: Homogenous Second Order Linear Differential Equations
Video - 4:35 Brief overview of second order DE's and quickly does 2 real roots example (one distinct, one repeated) Does not go into why solutions have the form that they do - MIT: Second Order Constant Coefficient Linear Differential Equations
Video / PDF Lectures and notes on second order ODE's covering all cases (real distinct, real repeated, complex) of roots and connection to harmonic oscillators - Paul's Notes: Real & Distinct Roots
Text Examples with real and distinct roots of the characteristic equation - Khan Academy: Linear Homogenous Equations 2
Video - 8:28 Example with two real distinct roots (finds general solution only) - Khan Academy: Linear Homogenous Equations 3
Video - 5:59 Introduces an initial condition to the previous example See 4th video in sequence for a second example that requires quadratic formula. - Paul's Notes: Complex Roots
Text Examples with complex roots of the characteristic equation - Khan Academy: Complex roots of the characteristic equation
Video - 10:27 Part 1 of 3 videos on complex roots case - PatrickJMT: Homogenous Second Order Linear DE: Complex Roots Example
Video - 3:40 Example with complex roots - Paul's Notes: Undetermined Coefficients
Text Using undetermined coefficients to solve a second order ODE - Khan Academy: Method of undetermined coefficients
Video - 10:11 Part 1 of 4 videos on undetermined coefficients - PatrickJMT: Method of Undetermined Coefficients Part 1
Video - 3:55 Part 1 of 2 videos on an undetermined coefficients example - MIT: Undetermined Coefficients
Video / PDF Course material on undetermined coefficients - Paul's Notes: Variation of Parameters
Text Using variation of parameters to solve a second order ODE - PatrickJMT: Variation of Parameters to solve a Differential Equation (second order)
Video - 11:03 Detailed example using variation of parameters - PatrickJMT: Variation of Parameters to solve a Differential Equation (second order) Ex 2
Video - 12:07 Second example using variation of parameters - Pat Plunkett Math Grad: Harmonic Oscillator
Interactive Allows you to enter in mass, damping, spring, and initial conditions, and then graphs position vs. time and position vs. velocity - Paul's Notes: Vibrations
Text Exampes involving springs - MIT: Damped Harmonic Oscillators
Video / PDF Course material on damped harmonic oscillators

Real Roots Case

Complex Roots Case

Undetermined Coefficients (2nd Order)

Variation of Parameters (2nd Order)

Oscillating Systems

- Paul's Notes: Higher Order Basic Concepts
Text Introduction to higher order ODEs - Paul's Notes: Higher Order Homogenous Differential Equations
Text Finding solutions to higher order homogenous ODEs - Paul's Notes: Higher Order Undetermined Coefficients
Text Using undetermined coefficients to solve a higher order ODE

- Paul's Notes: Systems of Differential Equations
Text Introduction to systems of differential equations - MIT: Linear Systems
Video / PDF Introduction to systems of differential equations and review material - MIT: Matrix Methods
Video / PDF Course material on real, complex, and repeated eigenvalue cases - Paul's Notes: Real Eigenvalues
Text Examples with real eigenvalues - Paul's Notes: Complex Eigenvalues
Text Examples with complex eigenvalues - Direction Field, n=2
Interactive Graphs direction fields for system of two differential equations - ODE Software for Matlab
Interactive Java version of Matlab direction field functions Use pplane for a system of first order ODEs - Paul's Notes: Phase Plane
Text Introduction to sketching phase planes - MIT: Phase Portraits
Video / PDF Course material on phase portraits

Phase Planes and Direction Fields