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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