Differential equations and classification
(0.2.1-2) Introduction to Differential Equations and Solutions to Differential Equations
Introduction to Differential Equation Terminology
(0.2.3) Differential Equations in Practice
(0.2.4) Four Fundamental Differential Equations and Their Solutions
(0.3) Lesson: Classifying Differential Equations
Differential Equations: Find the Order and Classify as Linear or Nonlinear
(0.3.101) Exercise 0.3.101: Classifying Differential Equations
Introduction to Differential Equations
Differential Equations and Exponential Functions
Standard and Differential Form of First-Order Differential Equations
Solutions and initial value problems
Intro to Initial Value Problems
Verifying Solutions to Differential Equations
Ex: Given a Solution to a Differential Equation, Find the Particular Solution
Ex 1: Verify a Solution to a Differential Equation, Find a Particular Solution
Ex 2: Verify a Solution to a Differential Equation, Find a Particular Solution
Ex: Determine Which Functions Are Solutions to a Differential Equation
Ex: Determine Which Function is a Solution to a Second Order Differential Equation
Ex: Verify a Solution to a Differential Equation and Find a Particular Solution
Ex: Find a Constant Function Solution to a Differential Equation
Direct integration
(1.1) Solutions to Differential Equations as Integrals: Form y'(x)=f(x)
(Example 1.1.5) Differential Equation App in the Form y'(x)=f(x): Speed = x'(t)=e^(t/2)
(Ex 1.1.2) Solve a Differential Equation in the Form y'(x)=f(x): y'=x^2+x,y(1)=3
(Ex 1.1.3) Solve a Differential Equation in the Form y'(x)=f(x): y'=sin(5x),y(0)=2
Ex: Solve a Basic Initial Value Problem (Linear)
Ex: Solve a Basic Initial Value Problem (Exponential and Trig)
Ex: Find the Particular Solution to a Basic Differential Equation
Ex: Initial Value Problem in the Form y' = kx Using Shortcut
Slope fields
Introduction to Slope Fields (1.2.1)
Create Slope Fields Using GeoGebra (1.2.1-3)
Ex: Determine Which Differential Equation Would Produce a Given Direction Field
Ex: Determine Direction Field Given a Solution to a Differential Equation
Ex: Select a Direction Field Given a Differential Equation Using Points
Existence uniqueness and intervals of validity
Existence and Uniqueness of y'=f(x,y) with y(x_0)=y_0: Picard's Theorem (1.2.2)
Find the Interval That a Linear First Order Differential Equation Has a Unique Solution
Find the Region that a First Order Differential Equation Has a Unique Solution at a Point - Part 1
Find Two Solutions to a First Order Initial Value Problem
Find Values Excluded to Guarantee Existence and Uniqueness of Solution to a IVP - y'=f(t,y)
Use Picard's Iteration to Approximate a Solution to a IVP (2 iterations only)
Ex: Existence and Uniqueness of y'=f(x,y) with y(x_0)=y_0: Picard's Theorem (1.2.102-103)
Ex 1: Find the Interval that Guarantees a Solution to an IVP Exists (Interval of Validity)
Ex 2: Find the Interval that Guarantees a Solution to an IVP Exists (Interval of Validity)
Separation of variables
(1.3.1) Introduction to Separation of Variables to Solve y'=f(x,y)
(1.3.2) Introduction to Separation of Variables to Solve y'=f(x,y) with an Implicit Solution
Differential Equations: Separation of Variables
Solving a differential equation by separation of variables
(Example 1.1.3) Solve a Differential Equation in the Form y'(x)=f(x): y'=ky
(Ex 1.1.5) Solve a Differential Equation in the Form y'(x)=f(y): y'=y^3,y(0)=1
(Ex 1.1.102) Solve a Differential Equation in the Form x'=f(x): x'=1/x^2 ,x(1)=2
(Example 1.3.2) Solve a Separable Differential Equation Using Factor by Grouping
(Example 1.3.4) Solve a Separable Differential Equation: y'=(-xy^2)/3
(Ex 1.3.102) Solve a IVP with a Separable Differential Equation: x'=3xt^2-3t^2
(Ex 1.4.104) Solve an IVP with a Separable Differential Equation: Slowing Growth Rate
Ex 1: Initial Value Problem Using Separation of Variables (Square Root)
Ex 2: Initial Value Problem Using Separation of Variables (Square Root)
Ex 1: Initial Value Problem Using Separation of Variables Involving Natural Logarithm
Ex 2: Initial Value Problem Using Separation of Variables Involving Natural Logarithm
Ex 1: IVP - Separation of Variables
Ex 2: IVP - Separation of Variables
Ex 1: Initial Value Problem Using Separation of Variables in the form y' = ky
Ex 2: Initial Value Problem Using Separation of Variables in the Form y'=ay+b
Ex: Initial Value Problem Using Separation of Variables in the Form y' = e^(ay+bx)
Ex: Solve an IVP Using Separation of Variables in the Form y'=(ax+b)/(xy^2)
Ex: Solve an IVP Using Separation of Variables in the Form y'=axy+bx
Ex: Solve an IVP with a Separable Differential Equation in the form y'=axy-bx
Ex: Solve an IVP with a Separable Differential Equation in the form dy/dt=t/(t^2y+y)
Ex: Solve an IVP with a Separable Differential Equation in the form (y^6x)dy/dx=1+x
Linear equations and integrating factors
(1.4) Introduction to Solving a First Order Linear Differential Equation Using an Integrating Factor
Solving Linear First-Order Differential Equations (integrating factor)
Solve a Linear First-Order DE Using an Integrating Factor
(Ex 1.4.1) Solve a IVP with a Linear Diff Equation Using an Integrating Factor: y'+2xy=e^(x-x^2)
(Ex 1.4.101) Solve a First Order Linear DE Using an Integrating Factor and Separation of Variables
(Ex 1.4.102) Solve a IVP with a Lin Differential Eq - Integrating Factor: y'+2sin(2x)y=2sin(2x)
Ex 1: Solve a Linear First-Order Differential Equation (integrating factor)
Ex 2: Solve a Linear First-Order Differential Equation (integrating factor)
Ex 3: Solve a Linear First-Order Differential Equation (IVP) – integrating factor
Ex 1: Solving a Linear First Order DE Using an Integrating Factor
Ex 2: Solve a Linear First Order DE Using an Integrating Factor
Ex 3: Solve a Linear First Order DE Using an Integrating Factor
Ex 4: Solve a Linear First Order DE Using an Integrating Factor
Ex 5: Solve a Linear First Order DE Using an Integrating Factor (Integration by Parts)
Ex 6: Solve a Linear First Order DE Using an Integrating Factor (Integration by Parts)
Ex: Solve a IVP Given Linear First Order DE Using an Integrating Factor
General substitutions
(1.5.1) Introduction to Solving Differential Equations Using Substitution
(Ex 1.5.104) Solve an First Order Differential Equation Using a General Substitution
(Ex 1.5) Solve an Initial Value Problem Using a General Substitution
Bernoulli equations
(1.5.2)Introduction to Solving Bernoulli Differential Equations
Solve a Bernoulli Differential Equation (Part 1)
Solve a Bernoulli Differential Equation (Part 2)
Solve a Bernoulli Differential Equation Initial Value Problem (Part 3)
(Ex 1.5.101) Solve an Initial Value Problem with a Bernoulli Differential Equation
(Ex 1.5.103) Solve an Initial Value Problem with a Bernoulli Differential Equation
Ex: Solve a Bernoulli Differential Equation Using Separation of Variables
Ex: Solve a Bernoulli Differential Equation Using an Integrating Factor
Homogeneous first order equations
(1.5.3)Introduction to Solving First Order Homogeneous Differential Equations
Determine if a Function is a Homogeneous Function
Determine if a First-Order Differential Equation is Homogeneous - Part 1
Determine if a First-Order Differential Equation is Homogeneous - Part 2
Solve a First-Order Homogeneous Differential Equation - Part 1
Solve a First-Order Homogeneous Differential Equation - Part 2
Solve a First-Order Homogeneous Differential Equation - Part 3
Solve a First-Order Homogeneous Differential Equation in Differential Form - Part 1
Solve a First-Order Homogeneous Differential Equation in Differential Form - Part 2
Solve a First-Order Homogeneous Differential Equation in Differential Form - Part 3
(Ex 1.5.102) Solve an Initial Value Problem with a Homogeneous Differential Equation
(Ex 1.5) Solve an Initial Value Problem with a Homogeneous Differential Equation
Exact equations and integrating factors
(1.8) Introduction to Solving Exact Differential Equations
Exact First Order Differential Equations - Part 1
Exact First Order Differential Equations - Part 2
(1.8.101A) Solve an Exact Differential Equation: cos(x)+ye^(xy)+xe^(xy)y'=0
(1.8.101B) Solve an Exact Differential Equation: (2x+y)dx+(x-4y)dy=0
(1.8.101C) Solve an Exact Differential Equation: e^x+e^y*y'=0
(1.8.101D) Solve an Exact Differential Equation: (3x^2+y)dx+(3y^2+3x)dy=0
(1.8.2) Solve a Differential Equation Using an Integrating Factor to form an Exact Equation
(Ex 1.8.7) Solve a Differential Equation Using an Integrating Factor to form an Exact Equation
(Ex 1.8.102B) Solve a Differential Equation Using an Integrating Factor to form an Exact Equation
(Ex 1.8.102C) Solve a Differential Equation Using an Integrating Factor to form an Exact Equation
Ex 1: Solve an Exact Differential Equation
Ex 2: Solve an Exact Differential Equation
Ex 3: Solve an Exact Differential Equation
Ex 4: Solve an Exact Differential Equation
Exponential growth decay and cooling
Applications of First Order Differential Equations - Exponential Growth: Part 1
Applications of First Order Differential Equations - Exponential Growth: Part 2
Applications of First Order Differential Equations: Exponential Decay Part 1
Applications of First Order Differential Equations - Exponential Decay Part 2
Applications of First Order Differential Equations - Newton's Law of Cooling
(Example 1.3.3) Solve Separable Differential Equation Involving Newton's Law of Cooling
Mixing and financial models
Applications of First Order Differential Equations - Mixing Concentrations
Applications of First Order Differential Equations - Mixing Concentrations 2
Separable Differential Equation Application: Mixture with Flow In/Out Equal
Separable Differential Equation Application: Mixture with Flow In/Out Different
Write a Differential Equation to Model the Change in a Bank Account: Changing Deposit Amount
(Example 1.4.2) Amount of Salt in a Mixing Tank by Solving a DE Using an Integrating Factor
Ex: Write a Differential Equation to Model the Change in a Bank Account
Ex: Solve a Differential Equation that Models the Change in a Bank Account Balance
First order circuit and motion models
Applications of First Order Differential Equations – RL Circuit
Applications of First Order Differential Equations – Falling Object
Separable Differential Equation Application: Kirchhoff's Law
Autonomous equations equilibrium and logistic growth
(1.6) Introduction to Autonomous Differential Equations
Find the Equilibrium Population and When the a Population is Increasing Given a Autonomous DE
(Ex 1.6.101) Create a Phase Diagram for an Autonomous Differential Equation
(Ex 1.6.102) Determine Any Critical Points and a Limit of an Autonomous Diff Equation: dx/dt=e^(-x)
Ex: Limited Growth Differential Equation
Ex 1: Solve an Autonomous DE IVP - Logistic Growth Using Separation of Variables
Ex 1: Solve an Autonomous DE IVP - Logistic Growth (Shortcut)
Ex 2: Solve an Autonomous DE IVP - Logistic Growth Using Separation of Variables
Write a Differential Equation to Model Logistic Sales Growth
(Ex 1.6.103) Determine a Diff Equation to Model a Fish Population and Determine the Limiting Pop
Numerical methods
(1.7) Introduction to Euler's Method to Approximate a Solution to an Initial Value Problem
(1.7.1) Use Euler's Method to Approximate a Function Value to the Solution of an Initial Value Prob
Approximate a Solution to a DE Using Euler's Method
Approximate a Function Value of a Solution to a DE Using Euler's Method
(1.7.2) Use the Improved Euler's Method to Approximate a Solution Function Value
4th Order Runge-Kutta Method for Approximating a Solution to a First Order IVP
Superposition linear independence and fundamental solutions
(2.1) Intro to Second Order Linear ODEs: Superposition, Existence/Uniqueness/Linear Independence
(2.1) Determine if Two Functions are Linearly Dependent or Linearly Independent
D-Notation and Determine a Linear Differential Operator of a Linear Differential Equation
Verify a Fundamental Set of Solutions for a Linear Second Order Homogeneous DE
Reduction of order and missing dependent variables
Reduction of Order - Linear Second Order Homogeneous Differential Equations Part 1
Reduction of Order - Linear Second Order Homogeneous Differential Equations Part 2
Shortcut Reduction of Order - Linear Second Order Homogeneous Differential Equations Part 1
Shortcut Reduction of Order - Linear Second Order Homogeneous Differential Equations Part 2
(2.1.104) Find a General Solution to a 2nd Order Linear Homogenous DE (Missing y Term)
(2.1.5) Find a General Solution to a 2nd Order Linear Homogenous DE (Missing y Term)
(Ex 2.1B): Find a General Solution to a 2nd Order Linear Homogenous DE (Reduction of Order)
(Ex 2.1B): Find a General Solution to a 2nd Order Linear Homogenous DE (Shortcut)
(Ex 2.1B2): Find a General Solution to a 2nd Order Linear Homogenous DE (Reduction of Order)
(Ex 2.1B2): Find a General Solution to a 2nd Order Linear Homogenous DE (Shortcut)
Second order equations with constant coefficients
(2.2) Solve 2nd Order Linear Homogeneous ODEs with Constant Coefficients
Linear Second Order Homogeneous Differential Equations - (two distict real roots)
Ex: Linear Second Order Homogeneous Differential Equations - (two distict real roots)
Ex: Solve and Verify the Solution of a Linear Second Order Homogeneous Differential Equation
Ex 1: Solve a Linear Second Order Homogeneous Differential Equation Initial Value Problem
Ex 2: Solve a Linear Second Order Homogeneous Differential Equation Initial Value Problem
Ex: Linear Second Order Homogeneous Differential Equations - (two real irrational roots)
Linear Second Order Homogeneous Differential Equations - (two real equal roots)
Ex: Linear Second Order Homogeneous Differential Equations - (two real equal roots)
Ex: Solve a Linear Second Order Homogeneous Differential Equation Initial Value Problem (equal)
Linear Second Order Homogeneous Differential Equations - (complex roots)
Ex: Linear Second Order Homogeneous Differential Equations - (complex roots)
Ex: Solve a Linear Second Order Homogeneous Differential Equation Initial Value Problem (complex)
Ex: Solve Linear Second Order Homogeneous DE IVP with Constant Coefficients (Complex)
Describe the End Behavior of the Solutions to a Linear Second Order Homogeneous DE
Homogeneous Cauchy Euler equations
Derive the Auxiliary Equation for a Cauchy-Euler Equation
General Solution to a Second Order Homogeneous Cauchy-Euler Equation (distinct real)
General Solution to a Second Order Homogeneous Cauchy-Euler Equation (equal roots)
General Solution to a Second Order Homogeneous Cauchy-Euler Equation (complex)
Cauchy-Euler Initial Value Problem
Ex: Solve a Second Order Cauchy-Euler DE Initial Value Problem (2 distinct real)
Ex: Solve a Second Order Cauchy-Euler DE Initial Value Problem (2 equal real)
Ex: Solve a Second Order Cauchy-Euler DE Initial Value Problem (Complex)
Higher order equations with constant coefficients
(2.3.1) Introduction to Higher Order Linear Differential Equations and Related Theorem
(2.3.104) Determine if 4 Functions Are Linearly Independent or Linearly Dependent
(2.3.2) Intro to Higher Order Linear Homogeneous Differential Equations wit Constant Coefficients
(2.3.1) Solve a 3rd Order Linear Homogeneous ODEs with Constant Coefficients Complex
(2.3.3) Solve a 3rd Order Linear Homogeneous ODEs with Constant Coefficients - Complex
(2.3.4) Given a Characteristic Equation, Find the 3rd Order DE and General Solutions
(2.3.102) Given a Roots of a Characteristic Equation, Find the 3rd Order DE and General Solutions
(2.3.2) Solve a 4th Order Linear Homogeneous ODEs with Constant Coefficients - Repeated
(2.3.101) Solve a 5th Order Linear Homogeneous ODEs with Constant Coefficients Repeated
Nonhomogeneous equations and undetermined coefficients
Prove the Form of the General Solution to a Linear Second Order Nonhomogeneous DE
The Form of the Particular Solution Using the Method of Undetermined Coefficients - Part 1
The Form of the Particular Solution Using the Method of Undetermined Coefficients - Part 2
Method of Undetermined Coefficients to Find a Particular Solution (trig)
Initial Value Problem Using Method of Undetermined Coefficients
Find a General Solution to a Nonhomogeneous DE Using Undetermined Coefficients (Linear)
Find a General Solution to a Nonhomogeneous DE Using Undetermined Coefficients (Quadratic)
Find a General Solution to a Nonhomogeneous DE Using Undetermined Coefficients (Exponential)
Find a General Solution to a Nonhomogeneous DE Using Undetermined Coefficients (Repeat Term)
Find a General Solution to a Nonhomogeneous DE Using Undetermined Coefficients (Sine)
Ex 1: Method of Undetermined Coefficients to Find the General Solution (exponential)
Ex 2: Method of Undetermined Coefficients to Find the General Solution (quadratic)
Variation of parameters
Derive the Variation of Parameters Formula to Solve Linear Second Order Nonhomogeneous DEs
Determine a Particular Solution of a Second Order DE using Variation of Parameters
Find a Particular Solution to a Nonhomgeneous DE Using Variation of Parameters
Ex 1: General Solution to a Second Order DE Using Variation of Parameters
Ex 2: General Solution to a Second Order DE Using Variation of Parameters
Ex 3: General Solution to a Second Order DE Using Variation of Parameters (trig)
Nonhomogeneous Cauchy Euler equations
Second Order Nonhomogeneous Cauchy-Euler Differential Equations
Free vibrations and circuit models
(2.4.1) Introduction to Mechanical Vibrations and Related Applications
(2.4.2) Introduction to Free Undamped Motion
Introduction to Free Undamped Motion (Spring System)
Ex 1: Free Undamped Motion IVP Problem (Spring System)
Ex 2: Free Undamped Motion IVP Problem (Spring System)
Ex 3: Free Undamped Motion IVP Problem (Spring System)
(2.4.101) Determine a Spring Constant k: Free Undamped Motion (Spring System)
(2.4.103) Free Undamped Motion: Train Hits a Bumper - Find k and Max Mass Under Various Conditions
(2.4.3) Introduction to Free Damped Motion: Overdamping, Critical Damping, Underdamping
Ex: Model Free Damped Vibration and Find Displacement Function
Ex: Determine a Dampening Force For An Overdamped System (Free Damped Vibration)
Ex: Determine a Dampening Force For An Critically Damped System (Free Damped Vibration)
(2.4.102) Setup and Solve an IVP for an RLC Circuit - Free Damped Motion, Underdamping
Forced vibrations and resonance
(2.6.1) Undamped Forced Motion and Resonance
(2.6.2A) Damped Forced Motion and Practical Resonance
(2.6.2B) Damped Forced Motion: Transient Solution and Steady Periodic Solution
Undamped Forced Motion: Mass on a Spring (General Solution Formula: w0 not equal w1)
Undamped Forced Motion: Mass on a Spring (General Solution Formula: w0 equal w1)
Damped Formed Motion: Mass on a Spring (General Solution Formula)
Damped Formed Motion: Mass on a Spring (Characteristic Equation and Undetermined Coefficients)
Definitions existence and basic transforms
(6.1.1) Introduction to the Laplace Transform
Introduction to Laplace Transforms
(6.1.2) The Existence and Uniqueness of a Laplace Transform
Ex: Find the Laplace Transform of f(t)=3 Using Definition
Ex: Find the Laplace Transform of f(t)=e^(2t) Using Definition
Ex: Find the Laplace Transform of f(t)=t^3 Using Definition
Ex: Find the Laplace Transform of f(t)=sin(6t) Using Definition
(6.1.101) Find a Laplace Transform: Quadratic Function
Inverse transforms and partial fractions
(6.1.3) Introduction to Inverse Laplace Transforms
Find Basic Inverse Laplace Transforms of the Form t^n
Find Inverse Laplace Transforms: e^(at) and t^n
Find Inverse Laplace Transforms: sin(at) and cos(at)
Find Inverse Laplace Transforms: sinh(at) and e^(at)
(6.1.102: Find an Inverse Laplace Transform Using Partial Fraction Decomposition
Ex 1: Find the Inverse Laplace Transform of Y(s) Using Partial Fractions
Ex 2: Find the Inverse Laplace Transform of Y(s) Using Partial Fractions
(6.3.102) Determine an Inverse Laplace Transform Using Partial Fractions
Transforms of derivatives and initial value problems
(6.2.1) Laplace Transforms of Derivatives / Using Laplace Transforms to Solve Differential Equations
Laplace Transform: Find Y(s)=L(y) Given a Homogeneous Differential Equation
Laplace Transform: Find Y(s)=L(y) Given a Nonhomogeneous Differential Equation
Ex 1: Solve a Homogeneous DE IVP Using Laplace Transforms
Ex 2: Solve a Homogeneous DE IVP Using Laplace Transforms
Unit step functions and piecewise forcing
Ex 1: Write a Step Function Using the Unit Step Function
Ex 2: Write a Step Function Using the Unit Step Function
Ex: Write a Ramp Function Using the Unit Step Function
(6.2.101) Define a Piecewise Function using Heaviside Functions (Unit Step Functions)
(6.1) The LaPlace Transform of the Unit Step Function
Ex: Find the Laplace Transform of a Step Function (method #1)
Ex: Find the Laplace Transform of a Step Function (method #2)
Ex: Find the Laplace Transform of a Piecewise Defined Function (Ramp)
Ex: Find the Laplace Transform of a f(t) Times a Unit Step Function
Determine a Laplace Transform Involving the a Heaviside Function (Ex 1)
Determine a Laplace Transform Involving the a Heaviside Function (Ex 2)
Determine a Laplace Transform Involving the a Heaviside Function (Ex 3)
(6.2.3) Laplace Transforms: Solving Differential Equations containing Heaviside Functions
(6.2.102) Solve an ODE with a Heaviside Function using Laplace Transforms
Transfer functions
(6.2.4) Laplace Transforms: Determining a Transfer Function
(6.2.103) Determine the Transfer Function of a First Order Differential Equation
Convolution and integral equations
(6.3.1) Intro to the Convolution and Using the Convolution to Find an Inverse Laplace Transform
(6.3.2) Introduction to Using a Convolution to Solve an Initial Value Problem
(6.3.3) Solving a Volterra Integral Equation
(6.3.101) Find the Convolution of Two Functions (Integration by Parts Twice)
(6.3.102) Determine an Inverse Laplace Transform Using Convolution
(6.3.103) Solve an Initial Value Problem Using Convolution
Ex: Find the Convolution of Two Exponential Functions
Ex: Find the Laplace Transform of the Convolution Integral
Ex: Find the Inverse Laplace Transform of the Convolution of Two Functions
Power series and Taylor series review
(7.1.1-2) Introduction to Power Series and Intervals of Convergence
(7.1.101) Determine if a Power Series is Convergent or Divergent
Ex 1: Interval of Convergence for Power Series (Centered at 0)
Ex 2: Interval of Convergence for Power Series (Centered at 0)
Ex 3: Interval of Convergence for Power Series (Centered at 0)
Ex 4: Interval of Convergence for Power Series (Centered at 0)
Ex 5: Interval of Convergence for Power Series (Not Centered at 0)
Ex 6: Interval of Convergence for Power Series (Not Centered at 0)
(7.1.4) Manipulation Power Series: Differentiation, Addition, and Multiplication
Differentiation and Integration Using Power Series
(7.1.5) Introduction to Power Series of Rational Functions
(7.1.103) Determine the Geometric Power Series with a Center of 2
Ex 1: Find a Power Series to Represent a Rational Function
Ex 2: Find a Power Series to Represent a Rational Function
Determine the Maclaurin Series and Polynomial for Function in the Form a*cos(bx^2)
Ex: Determine a Taylor Polynomial for a Square Root Function
Ex: Find a Maclaurin Polynomial and Error of an Approximation - ln(cos(x))
Taylor and power series solutions of differential equations
Taylor Series Solutions to Initial Value Problems - 1st Order
Taylor Series Solutions to Initial Value Problems - 2nd Order
(7.2.1A) Power Series Solutions to Second Order Linear ODEs: y''-y=0
(7.2.1B) Power Series Solutions to Second Order Linear ODEs: y''-xy=1 (Airy's Equation)
(7.2.1C) Power Series Solution to Hermite's Equation: y''-2xy'+2ny=0
7.2.101 Power Series Solution to a Second Order Linear OER: y''+2x^3y=0 (Term by Term)
Systems modeling and conversion to first order
(3.1.1) Introduction to Systems of Ordinary Differential Equations
(3.1.2) Introduction to Modeling Applications with Systems of Ordinary Differential Equations
(3.1.4) Introduction to Autonomous Systems of ODEs and Phase Portraits
(3.1.3) Changing Higher Order ODEs and Systems of ODEs as First Order Systems
(3.1.101) Solve a Basic System of Differential Equations (One at a Time)
(3.1.102) Solve a Basic System of Differential Equations (Substitution)
(3.1.103) Write a 3rd Order ODE as a 1st Order System of ODEs
(3.1.104) Write a 2nd Order System of ODEs as a 1st Order System of ODEs
Matrix algebra review
(3.2.1) Matrices, Matrix Properties, and Basic Matrix Operations
(3.2.2) Matrix Multiplication, Multiplication Properties, and Inverse Matrices
(3.2.3) The Determinant of Square Matrices and Properties
Ex 1: Determinant of 3x3 Matrix - Cofactor Method
Ex 2: Determinant of 3x3 Matrix - Cofactor Method
The Determinant of a 4 by 4 Matrix Using Cofactor Expansion (Expansion by Minors)
(3.2.4A) Solving a System of Linear Equations Using an Augmented Matrix
Ex 1: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)
Ex 2: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)
Solve a System of 3 Equations with 3 Unknowns Using an Augmented Matrix (RREF - 1 Solution)
Ex: Solve a System of Equation By Writing an Augmented Matrix in RREF (3x3 Infinite Sol)
(3.2.4B) Solving a System of Linear Equations Using a Matrix Equation
Ex: Inverse of a 2x2 Matrix Using an Augmented Matrix
Ex 1: Inverse of a 3x3 Matrix Using an Augmented Matrix
Ex 2: Inverse of a 3x3 Matrix Using an Augmented Matrix
Ex: Solve the Matrix Equation AX=B (3x3)
Ex: Solve a System of Three Equations Using a Matrix Equation
(Ex 3.2.6) Determine an Unknown Entry that Makes a 3 by 3 Matrix Not Invertible
(Ex 3.2.7) Determine an Unknown Entry that Makes a 3 by 3 Matrix Not Invertible
(Ex 3.2.104) Find the Inverses of a 2 by 2 Matrix and 3 by 3 Matrix with Variable Entries
Matrix notation and homogeneous system theory
(3.3.1) Intro to Vector-Valued and Matrix-Valued Functions and Write a System of ODEs Using Matrices
(3.3.2) Intro to Solving First Order Linear Homogeneous Systems of Ordinary Differential Equations
(3.3.103) Write a System of Two Differential Equations Using Matrix Notation
(Ex 3.3.1A) Write a System of Two Differential Equations with Initial Values Using Matrix Notation
(Ex 3.3.1B) Write a System of Three Differential Equations with Initial Values Using Matrices
(Ex 3.3A) Determine if Vector-Valued Functions are Linearly Independent or Linearly Dependent Video A | Video B
(Ex 3.3.102) Determine if Vector-Valued Functions are Linearly Independent or Linearly Dependent
(Ex 3.3.102) Write a System Linear First Order System of ODEs Using Matrix Notation and Solve
Eigenvalues eigenvectors and distinct real eigenvalues
(3.4.1) Intro to Solving a Lin Sys of ODEs using the Eigenvalue Method: Eigenvalues and Eigenvectors
(3.4.2) Solve a Linear System of ODEs using the Eigenvalue Method: Real, Distinct Eigenvalues
(3.4.101) Find Eigenvalues, Eigenvectors, and Solve a Lin Sys of ODEs with Const Coef: Homogeneous
(3.4.103) Solve a Linear System of ODEs using the Eigenvalue Method: Real, Distinct Eigenvalues
Complex eigenvalues
(3.4.3) Solve a Linear System of ODEs using the Eigenvalue Method: Complex Eigenvalues
(3.4.104) Solve a Linear System of ODEs using the Eigenvalue Method: Imaginary Eigenvalues
(3.4.102) Solve a Linear System of ODEs using the Eigenvalue Method: Complex Eigenvalues
Repeated eigenvalues
(3.7) Intro to Solving a Linear System of ODEs using the Eigenvalue Method: Repeated Eigenvalues
(3.7.2) Solve a Linear System of ODEs using the Eigenvalue Method: Repeated Eigenvalues, 1 Defect
(3.7.101) Solve a Linear System of ODEs using the Eigenvalue Method: Repeat Eigenvalues, No Defects
(3.7.102) Solve a Linear System of ODEs using the Eigenvalue Method: Repeated Eigenvalues, 1 Defect
Phase portraits and behavior of linear systems
(3.5) Types of Behavior of 2D Linear Homogeneous Const Coefficient Systems of ODEs Using Eigenvalues
(3.5.101) Describe the Behavior of 2D Lin Homogeneous Const Coefficient Systems of ODEs: Eigenvalues
Matrix exponentials
(3.8.1) Intro to Matrix Exponentials and Determining Matrix Exponentials for Simple Cases
(3.8.2) Matrix Exponentials for the General Case: N by N with N Linearly Indep Eigenvectors
(3.8.101) Determine a Matrix Exponential and General Solution to x'=Ax: 2 by 2 General Case
(3.8.102) Determine a Matrix Exponential and General Solution to x'=Ax: 3 by 3 General Case
(3.8.104) Approximate a Matrix Exponential Using the Taylor Expansion: 2 by 2
Second order systems and coupled spring applications
(3.6) Intro to Undamped Mass-Spring Systems: Second Order Systems of Ordinary Diff Equations
(3.6) Intro to Undamped Mass-Spring Systems with Periodic Forcing: 2nd Order Systems of ODEs
(3.6.1) Set-up and Solve an Undamped Mass Spring System: Second Order System of ODEs
(3.6.101) Solve a Linear Second Order Nonhomogeneous System of ODEs
Phase planes critical points and linearization
(8.1) A General Approach to Nonlinear Differential Questions
(8.1.1) Systems of Autonomous Nonlinear Differential Equations and Phase Plane Analysis
(8.1.2) Critical Points and the Linearization of a Nonlinear Autonomous System of ODEs
(8.1.101A) Critical Points and the Linearization of a Nonlinear Autonomous System of ODEs
(8.1.101B) Critical Points and the Linearization of a Nonlinear Autonomous System of ODEs
(8.1.101C) Critical Points and the Linearization of a Nonlinear Autonomous System of ODEs
(8.1.102) Match Nonlinear Autonomous Systems ODEs to Vector Fields (Geogebra)
(8.1.102) Creating Phase Portraits for Nonlinear Autonomous Systems of ODEs
Stability and almost linear systems
(8.2.1) Isolated Critical Points and Almost Linear Systems of Nonlinear Systems of ODEs
(8.2.2) Behavior and Stability of Isolated Critical Points of Nonlinear Systems of ODEs
(8.2.3) The Trouble with Behavior of Centers of Nonlinear Systems of ODEs
(8.2.101B) Find and Classify the Critical Points of a Nonlinear System of ODEs
(8.2.101C) Find and Classify the Critical Points of a Nonlinear System of ODEs
Conservative equations and trajectories
(8.2.4) Conversative 2nd Order ODEs and the Hamiltonian to Find Trajectories
(8.2.102A) The Equations of Trajectories and Classify the Critical Points of a Conservative Equation
(8.2.102B) The Equations of Trajectories and Classify the Critical Points of a Conservative Equation
(8.2.102C) The Equations of Trajectories and Classify the Critical Points of a Conservative Equation
Pendulum and predator prey applications
(8.3.1) Application of a Nonlinear System of ODEs: Trajectories of the Pendulum Equation
(8.3.2) Application of Nonlinear Systems of ODEs: Predator-Prey Model (Lotka-Volterra System)
Boundary conditions eigenvalues and eigenfunctions
Intro to Boundary Value Problems
(4.1.1): Boundary Value Problems
(4.1.2) Introduction to Finding Eigenvalues and Eigenfunction of Boundary Value Problems
(4.1.2B) Finding Eigenvalues and Eigenfunction of Boundary Value Problems
(4.1.2C) Finding Eigenvalues and Eigenfunction of Boundary Value Problems
(4.1.103) Solve a Boundary Value Problem
Orthogonality and the Fredholm alternative
(4.1.3) Orthogonality of Eigenfunctions Theorem and Proof
(4.1.4) The Fredholm Alternative
Spinning string applications
(4.1.5) Boundary Volume Problem Application - Spinning String
(4.1.101) Find Min Angular Velocity for a Spinning String to Pop Out: Boundary Value Application