Click to playlist as a pdf: Differential Equation

Introduction to Differential Equations 

Introduction to Differential Equation Terminology
Differential Equations:  Find the Order and Classify as Linear or Nonlinear
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
Verifying Solutions to Differential Equations
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
Ex: Determine Which Functions Are Solutions to a Differential Equation
Slope Fields
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
Intro to Initial Value Problems
Ex: Solve a Basic Initial Value Problem (Linear)
Ex: Solve a Basic Initial Value Problem (Exponential and Trig)
Intro to Boundary Value Problems
Introduction to Differential Equations
Differential Equations and Exponential Functions
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 the Region that a First Order Differential Equation Has a Unique Solution Through a Point - Part 2
Find Two Solutions to a First Order Initial Value Problem

First Order Differential Equations

Introduction to Differential Equations
Solving a differential equation by separation of variables
Ex:  Find the Particular Solution to a Basic Differential Equation
Standard and Differential Form of First-Order Differential Equations

Separation of Variables

Differential Equations:  Separation of Variables
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:  Separation of Variables
Ex 2:  Separation of Variables
Ex 3:  Separation of Variables
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 in the Form y' = kx Using Shortcut
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
Find Two Solutions to a First Order Initial Value Problem
Ex: Write a Differential Equation to Model the Change in a Bank Account
Ex: Limited Growth Differential Equation
Ex: Solve a Differential Equation that Models the Change in a Bank Account Balance

Exact First Order Differential Equations

Exact First Order Differential Equations - Part 1
Exact First Order Differential Equations - Part 2
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

Linear First Order Differential Equations

Integrating Factor Method

Solving Linear First-Order Differential Equations (integrating factor)
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 7: Solve a Linear First Order DE Using an Integrating Factor  (Integration by Parts Twice with Sub)
Ex: Solve a IVP Given Linear First Order DE Using an Integrating Factor

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: Solve a Bernoulli Differential Equation Using Separation of Variables
Ex: Solve a Bernoulli Differential Equation Using an Integrating Factor

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 Equatifon - 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

Interval of Validity (Existence and Uniqueness

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)
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)

Applications of First Order Differential Equations

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
Applications of First Order Differential Equations - Mixing Concentrations
Applications of First Order Differential Equations - Mixing Concentrations 2
Applications of First Order Differential Equations – RL Circuit
Applications of First Order Differential Equations – Falling Object
Separable Differential Equation Application:  Mixture with Flow In/Out Equal
Separable Differential Equation Application:  Mixture with Flow In/Out Different
Separable Differential Equation Application:  Kirchhoff's Law
Write a Differential Equation to Model the Change in a Bank Account: Changing Deposit Amount

Equilibrium Solutions

Find the Equilibrium Population and When the a Population is Increasing Given a Autonomous DE
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

Euler’s Method
Approximate a Solution to a DE Using Euler's Method
Approximate a Function Value of a Solution to a DE Using Euler's Method

Second Order Differential Equations

Linear Dependent Functions
Linear Independent Functions

Linear Second Order Homogeneous Differential Equations

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)
 Ex: Solve Linear Second Order Homogeneous DE with Constant Coefficients to Determine an Initial Condition
Describe the End Behavior of the Solutions to a Linear Second Order Homogeneous DE

Cauchy-Euler

Derive the Auxiliary Equation for a Cauchy-Euler Equation
Cauchy-Euler Initial Value Problem
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)
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)
Second Order Nonhomogeneous Cauchy-Euler Differential Equations

Reduction of Order

Verify a Fundamental Set of Solutions for a Linear Second Order Homogeneous DE
Reduction of Order - Linear Second Order Homogeneous Differential Equations Part 1
Shortcut 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 2

Method of 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)
Ex 1: Method of Undetermined Coefficients to Find the General Solution (exponential)
 Ex 2: Method of Undetermined Coefficients to Find the General Solution (quadratic)
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)

Variation of Parameters

Derive the Variation of Parameters Formula to Solve Linear Second Order Nonhomogeneous DEs
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)
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

Modeling with Higher Order Differential Equations

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)
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)

Laplace Transforms

Introduction to Laplace Transforms
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
Laplace Transform:  Find Y(s)=L(y) Given a Homogeneous Differential Equation
Laplace Transform:  Find Y(s)=L(y) Given a Nonhomogeneous Differential Equation
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)
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
Ex 1:  Solve a Homogeneous DE IVP Using Laplace Transforms
Ex 2:  Solve a Homogeneous DE IVP Using Laplace Transforms
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
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
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

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