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A University Level Introductory Course in Differential Equations

(1) Introduction to Differential Equations and their Applications

  • Object falling under the force of gravity
  • Object falling under the force of gravity and air resistance
  • Motion of a mass on a spring
  • RLC Circuits
  • Motion of a simple pendulum
  • More Differential Equation Models
  • Defining and Classifying Differential Equations
  • Solutions of Differential Equations
  • Explicit and Implicit Solutions
  • Slope Fields and Solution Curves
  • Existence and Uniqueness for first order Differential Equations
  • Separable Differential Equations
  • Separable Differential Equation Examples
  • Newtons Law of Cooling
  • Newtons Law of Cooling: Homicide Victim Example
  • Torricellis Law
  • Torricellis Law Example
  • Linear First Order Differential Equations
  • Differential Equation for Mixing Problems
  • Mixing Problem Example
  • Exact Differential Equations
  • Exact Differential Equation Example 1
  • Exact Differential Equation Example 2
  • Introduction to Substitution Methods
  • Homogenous Differential Equations
  • Homogeneous Differential Equation Example 1
  • Homogeneous Differential Equation Example 2
  • Bernoulli Differential Equations
  • Reduction of a Second Order Differential Equation to a First Order One
  • Assignment #1
  • Higher Order Differential Equations
  • Linear Differential Operators
  • Principal of Superposition
  • Existence and Uniqueness Theorem
  • The Wronskian Determinant
  • General Solutions of Second Order Linear Homogenous Equations
  • Summary of Theory for Second Order Homogenous Equations
  • Linear Independence and the Wronskian
  • Wronskian of Solutions
  • Theory of Higher Order Equations
  • Solving Second Order Equations with Constant Coefficients
  • Second Order Equations with Constant Coefficients: Distinct Roots
  • Solving Second Order Equations with Constant Coefficients: 1 Root
  • Solving Second Order Equations with Constant Coefficients: Complex Roots
  • Method of Reduction
  • Higher Order Equations: Distinct Real Roots
  • Higher Order Equations: Repeated Real Roots
  • Higher Order Equations: Distinct Complex Roots
  • Higher Order Equations: Repeated Complex Roots
  • Higher Order Equations: Example With All Cases
  • Nonhomogenous Differential Equations
  • Method of Undetermined Coefficients Example 1
  • Method of Undetermined Coefficients Example 2
  • Method of Undetermined Coefficients Example 3
  • Method of Undetermined Coefficients: Avoiding Duplication
  • Method of Undetermined Coefficients In General
  • Method of Undetermined Coefficients Example 4
  • Method of Undetermined Coefficients Example 5
  • Method of Undetermined Coefficients Example 6
  • Assignment 2
  • Reduction of Order: The General Formula
  • Reduction of Order: An Example
  • Variation of Parameters
  • Variation of Parameters: An Example
  • Assignment 3
  • The Laplace Transform
  • Laplace Transform Example: Unit Step Function
  • Laplace Transform Example: First Derivative
  • Laplace Transform Example: Second Derivative
  • Existence of the Laplace Transform
  • Laplace Transform Example: Exponential Function
  • Laplace Transform Example: Cosine, Sine, Hyperbolic Cosine and Sine
  • The Inverse Laplace Transform
  • Solving Differential Equations with Laplace Transform
  • Solving Differential Equations with Laplace Transform
  • Partial Fractions to Invert Transforms
  • First Translation Theorem
  • First Translation Theorem: Inverting Transforms
  • First Translation Theorem: Inverting Transforms: Completing Square
  • Second Translation Theorem
  • Piecewise Continuous Functions with Unit Step Functiond
  • Laplace Transform of Piecewise Continuous Functions
  • Laplace Transform of Piecewise Continuous Functions
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
  • Assignment 4
  • Derivatives of Transforms
  • Laplace Transform of Piecewise Periodic Functions
  • Solving an IVP with a Piecewise Periodic Non-homogenous Term
  • The Dirac Delta Function
  • Solving an IVP with a Delta Function Term
  • Solving an IVP with Multiple Delta Function Term
  • The Convolution Theorem
  • Convolution Theorem: Finding Integral Solutions
  • Convolution Theorem: Finding Integral Solutions
  • Assignment 5

(2) First Order Differential Equations

  • Object falling under the force of gravity
  • Object falling under the force of gravity and air resistance
  • Motion of a mass on a spring
  • RLC Circuits
  • Motion of a simple pendulum
  • More Differential Equation Models
  • Defining and Classifying Differential Equations
  • Solutions of Differential Equations
  • Explicit and Implicit Solutions
  • Slope Fields and Solution Curves
  • Existence and Uniqueness for first order Differential Equations
  • Separable Differential Equations
  • Separable Differential Equation Examples
  • Newtons Law of Cooling
  • Newtons Law of Cooling: Homicide Victim Example
  • Torricellis Law
  • Torricellis Law Example
  • Linear First Order Differential Equations
  • Differential Equation for Mixing Problems
  • Mixing Problem Example
  • Exact Differential Equations
  • Exact Differential Equation Example 1
  • Exact Differential Equation Example 2
  • Introduction to Substitution Methods
  • Homogenous Differential Equations
  • Homogeneous Differential Equation Example 1
  • Homogeneous Differential Equation Example 2
  • Bernoulli Differential Equations
  • Reduction of a Second Order Differential Equation to a First Order One
  • Assignment #1
  • Higher Order Differential Equations
  • Linear Differential Operators
  • Principal of Superposition
  • Existence and Uniqueness Theorem
  • The Wronskian Determinant
  • General Solutions of Second Order Linear Homogenous Equations
  • Summary of Theory for Second Order Homogenous Equations
  • Linear Independence and the Wronskian
  • Wronskian of Solutions
  • Theory of Higher Order Equations
  • Solving Second Order Equations with Constant Coefficients
  • Second Order Equations with Constant Coefficients: Distinct Roots
  • Solving Second Order Equations with Constant Coefficients: 1 Root
  • Solving Second Order Equations with Constant Coefficients: Complex Roots
  • Method of Reduction
  • Higher Order Equations: Distinct Real Roots
  • Higher Order Equations: Repeated Real Roots
  • Higher Order Equations: Distinct Complex Roots
  • Higher Order Equations: Repeated Complex Roots
  • Higher Order Equations: Example With All Cases
  • Nonhomogenous Differential Equations
  • Method of Undetermined Coefficients Example 1
  • Method of Undetermined Coefficients Example 2
  • Method of Undetermined Coefficients Example 3
  • Method of Undetermined Coefficients: Avoiding Duplication
  • Method of Undetermined Coefficients In General
  • Method of Undetermined Coefficients Example 4
  • Method of Undetermined Coefficients Example 5
  • Method of Undetermined Coefficients Example 6
  • Assignment 2
  • Reduction of Order: The General Formula
  • Reduction of Order: An Example
  • Variation of Parameters
  • Variation of Parameters: An Example
  • Assignment 3
  • The Laplace Transform
  • Laplace Transform Example: Unit Step Function
  • Laplace Transform Example: First Derivative
  • Laplace Transform Example: Second Derivative
  • Existence of the Laplace Transform
  • Laplace Transform Example: Exponential Function
  • Laplace Transform Example: Cosine, Sine, Hyperbolic Cosine and Sine
  • The Inverse Laplace Transform
  • Solving Differential Equations with Laplace Transform
  • Solving Differential Equations with Laplace Transform
  • Partial Fractions to Invert Transforms
  • First Translation Theorem
  • First Translation Theorem: Inverting Transforms
  • First Translation Theorem: Inverting Transforms: Completing Square
  • Second Translation Theorem
  • Piecewise Continuous Functions with Unit Step Functiond
  • Laplace Transform of Piecewise Continuous Functions
  • Laplace Transform of Piecewise Continuous Functions
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
  • Assignment 4
  • Derivatives of Transforms
  • Laplace Transform of Piecewise Periodic Functions
  • Solving an IVP with a Piecewise Periodic Non-homogenous Term
  • The Dirac Delta Function
  • Solving an IVP with a Delta Function Term
  • Solving an IVP with Multiple Delta Function Term
  • The Convolution Theorem
  • Convolution Theorem: Finding Integral Solutions
  • Convolution Theorem: Finding Integral Solutions
  • Assignment 5

(3) Higher Order Differential Equations

  • Object falling under the force of gravity
  • Object falling under the force of gravity and air resistance
  • Motion of a mass on a spring
  • RLC Circuits
  • Motion of a simple pendulum
  • More Differential Equation Models
  • Defining and Classifying Differential Equations
  • Solutions of Differential Equations
  • Explicit and Implicit Solutions
  • Slope Fields and Solution Curves
  • Existence and Uniqueness for first order Differential Equations
  • Separable Differential Equations
  • Separable Differential Equation Examples
  • Newtons Law of Cooling
  • Newtons Law of Cooling: Homicide Victim Example
  • Torricellis Law
  • Torricellis Law Example
  • Linear First Order Differential Equations
  • Differential Equation for Mixing Problems
  • Mixing Problem Example
  • Exact Differential Equations
  • Exact Differential Equation Example 1
  • Exact Differential Equation Example 2
  • Introduction to Substitution Methods
  • Homogenous Differential Equations
  • Homogeneous Differential Equation Example 1
  • Homogeneous Differential Equation Example 2
  • Bernoulli Differential Equations
  • Reduction of a Second Order Differential Equation to a First Order One
  • Assignment #1
  • Higher Order Differential Equations
  • Linear Differential Operators
  • Principal of Superposition
  • Existence and Uniqueness Theorem
  • The Wronskian Determinant
  • General Solutions of Second Order Linear Homogenous Equations
  • Summary of Theory for Second Order Homogenous Equations
  • Linear Independence and the Wronskian
  • Wronskian of Solutions
  • Theory of Higher Order Equations
  • Solving Second Order Equations with Constant Coefficients
  • Second Order Equations with Constant Coefficients: Distinct Roots
  • Solving Second Order Equations with Constant Coefficients: 1 Root
  • Solving Second Order Equations with Constant Coefficients: Complex Roots
  • Method of Reduction
  • Higher Order Equations: Distinct Real Roots
  • Higher Order Equations: Repeated Real Roots
  • Higher Order Equations: Distinct Complex Roots
  • Higher Order Equations: Repeated Complex Roots
  • Higher Order Equations: Example With All Cases
  • Nonhomogenous Differential Equations
  • Method of Undetermined Coefficients Example 1
  • Method of Undetermined Coefficients Example 2
  • Method of Undetermined Coefficients Example 3
  • Method of Undetermined Coefficients: Avoiding Duplication
  • Method of Undetermined Coefficients In General
  • Method of Undetermined Coefficients Example 4
  • Method of Undetermined Coefficients Example 5
  • Method of Undetermined Coefficients Example 6
  • Assignment 2
  • Reduction of Order: The General Formula
  • Reduction of Order: An Example
  • Variation of Parameters
  • Variation of Parameters: An Example
  • Assignment 3
  • The Laplace Transform
  • Laplace Transform Example: Unit Step Function
  • Laplace Transform Example: First Derivative
  • Laplace Transform Example: Second Derivative
  • Existence of the Laplace Transform
  • Laplace Transform Example: Exponential Function
  • Laplace Transform Example: Cosine, Sine, Hyperbolic Cosine and Sine
  • The Inverse Laplace Transform
  • Solving Differential Equations with Laplace Transform
  • Solving Differential Equations with Laplace Transform
  • Partial Fractions to Invert Transforms
  • First Translation Theorem
  • First Translation Theorem: Inverting Transforms
  • First Translation Theorem: Inverting Transforms: Completing Square
  • Second Translation Theorem
  • Piecewise Continuous Functions with Unit Step Functiond
  • Laplace Transform of Piecewise Continuous Functions
  • Laplace Transform of Piecewise Continuous Functions
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
  • Assignment 4
  • Derivatives of Transforms
  • Laplace Transform of Piecewise Periodic Functions
  • Solving an IVP with a Piecewise Periodic Non-homogenous Term
  • The Dirac Delta Function
  • Solving an IVP with a Delta Function Term
  • Solving an IVP with Multiple Delta Function Term
  • The Convolution Theorem
  • Convolution Theorem: Finding Integral Solutions
  • Convolution Theorem: Finding Integral Solutions
  • Assignment 5

(4) Laplace Transforms

  • Object falling under the force of gravity
  • Object falling under the force of gravity and air resistance
  • Motion of a mass on a spring
  • RLC Circuits
  • Motion of a simple pendulum
  • More Differential Equation Models
  • Defining and Classifying Differential Equations
  • Solutions of Differential Equations
  • Explicit and Implicit Solutions
  • Slope Fields and Solution Curves
  • Existence and Uniqueness for first order Differential Equations
  • Separable Differential Equations
  • Separable Differential Equation Examples
  • Newtons Law of Cooling
  • Newtons Law of Cooling: Homicide Victim Example
  • Torricellis Law
  • Torricellis Law Example
  • Linear First Order Differential Equations
  • Differential Equation for Mixing Problems
  • Mixing Problem Example
  • Exact Differential Equations
  • Exact Differential Equation Example 1
  • Exact Differential Equation Example 2
  • Introduction to Substitution Methods
  • Homogenous Differential Equations
  • Homogeneous Differential Equation Example 1
  • Homogeneous Differential Equation Example 2
  • Bernoulli Differential Equations
  • Reduction of a Second Order Differential Equation to a First Order One
  • Assignment #1
  • Higher Order Differential Equations
  • Linear Differential Operators
  • Principal of Superposition
  • Existence and Uniqueness Theorem
  • The Wronskian Determinant
  • General Solutions of Second Order Linear Homogenous Equations
  • Summary of Theory for Second Order Homogenous Equations
  • Linear Independence and the Wronskian
  • Wronskian of Solutions
  • Theory of Higher Order Equations
  • Solving Second Order Equations with Constant Coefficients
  • Second Order Equations with Constant Coefficients: Distinct Roots
  • Solving Second Order Equations with Constant Coefficients: 1 Root
  • Solving Second Order Equations with Constant Coefficients: Complex Roots
  • Method of Reduction
  • Higher Order Equations: Distinct Real Roots
  • Higher Order Equations: Repeated Real Roots
  • Higher Order Equations: Distinct Complex Roots
  • Higher Order Equations: Repeated Complex Roots
  • Higher Order Equations: Example With All Cases
  • Nonhomogenous Differential Equations
  • Method of Undetermined Coefficients Example 1
  • Method of Undetermined Coefficients Example 2
  • Method of Undetermined Coefficients Example 3
  • Method of Undetermined Coefficients: Avoiding Duplication
  • Method of Undetermined Coefficients In General
  • Method of Undetermined Coefficients Example 4
  • Method of Undetermined Coefficients Example 5
  • Method of Undetermined Coefficients Example 6
  • Assignment 2
  • Reduction of Order: The General Formula
  • Reduction of Order: An Example
  • Variation of Parameters
  • Variation of Parameters: An Example
  • Assignment 3
  • The Laplace Transform
  • Laplace Transform Example: Unit Step Function
  • Laplace Transform Example: First Derivative
  • Laplace Transform Example: Second Derivative
  • Existence of the Laplace Transform
  • Laplace Transform Example: Exponential Function
  • Laplace Transform Example: Cosine, Sine, Hyperbolic Cosine and Sine
  • The Inverse Laplace Transform
  • Solving Differential Equations with Laplace Transform
  • Solving Differential Equations with Laplace Transform
  • Partial Fractions to Invert Transforms
  • First Translation Theorem
  • First Translation Theorem: Inverting Transforms
  • First Translation Theorem: Inverting Transforms: Completing Square
  • Second Translation Theorem
  • Piecewise Continuous Functions with Unit Step Functiond
  • Laplace Transform of Piecewise Continuous Functions
  • Laplace Transform of Piecewise Continuous Functions
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
  • Assignment 4
  • Derivatives of Transforms
  • Laplace Transform of Piecewise Periodic Functions
  • Solving an IVP with a Piecewise Periodic Non-homogenous Term
  • The Dirac Delta Function
  • Solving an IVP with a Delta Function Term
  • Solving an IVP with Multiple Delta Function Term
  • The Convolution Theorem
  • Convolution Theorem: Finding Integral Solutions
  • Convolution Theorem: Finding Integral Solutions
  • Assignment 5

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