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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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