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Home / Given an Ode Find Its General Solution Patrickjmt

Given an Ode Find Its General Solution Patrickjmt

Online Math Lab

Math 4B
Differential Equations

  • 1st Order Differential Equations
    • Direction Fields
    • Separable Differential Equations
    • Linear Differential Equations
    • Applications
  • 2nd Order Differential Equations
    • Real Roots Case
    • Complex Roots Case
    • Undetermined Coefficients (2nd Order)
    • Variation of Parameters (2nd Order)
    • Oscillating Systems
  • Higher Order Differential Equations
  • Systems of 1st Order Differential Equations
    • Phase Planes and Direction Fields

1st Order 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

    • Direction Fields

    • 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

    Separable Differential Equations

    • 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

    Linear Differential Equations

    • 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

    Applications

    • 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

2nd Order Differential Equations

  • 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

    • Real Roots Case

    • 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.

    Complex Roots Case

    • 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

    Undetermined Coefficients (2nd Order)

    • 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

    Variation of Parameters (2nd Order)

    • 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

    Oscillating Systems

    • 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

Higher Order Differential Equations

  • 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

Systems of 1st Order Differential Equations

  • 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

    • Phase Planes and Direction Fields

    • 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

Given an Ode Find Its General Solution Patrickjmt

Source: http://web.math.ucsb.edu/oml/4B.html