**Second Order Differential Equations**

## Distinct Real Roots

41 min 5 Examples

- Overview of Second-Order Differential Equations with Distinct Real Roots
- Example – verify the Principal of Superposition
- Example #1 – find the General Form of the Second-Order DE
- Example #2 – solve the Second-Order DE given Initial Conditions
- Example #3 – solve the Second-Order DE given Initial Conditions
- Example #4 – find the General Form of the Second-Order DE
- Example #5 – find the General Form of the Second-Order DE

## Complex Roots

46 min 5 Examples

- Overview of Second-Order Differential Equations with Complex Roots
- Example #1 – find the General Form of the Second-Order DE
- Example #2 – find the General Form of the Second-Order DE
- Overview of Transient Terms and Steady-State Terms
- Example #3 – solve the Second-Order DE given Initial Conditions
- Example #4 – solve the Second-Order DE given Initial Conditions
- Example #5 – Non-Constant Coefficients and Intro to Reduction of Order

## Repeated Roots

1 hr 9 min 6 Examples

- Overview of Second-Order Differential Equations with Repeated Roots and Reduction of Order
- Example #1 – find the General Solution to the Second-Order DE
- Example #2 – solve the Second-Order DE given Initial Conditions
- Example #3 – solve the Second-Order DE given Initial Conditions
- Example #4 – Use Reduction of Order to find a Second Independent Solution
- Example #5 – Use Reduction of Order to find a Second Independent Solution
- Example #6 – Use Reduction of Order to find a Second Independent Solution
- Recap of How to Solve Second-Order Homogeneous DEs and Reduction of Order

## Wronskian

45 min 6 Examples

- Overview of Differential Operators and Review the Principle of Superposition
- Overview of the Wronskian and Important Theorems and Facts
- Example #1 – calculate the Wronskian for a 2rd-Order DE and determine if Linearly Independent
- Example #2 – calculate the Wronskian for a 2rd-Order DE and determine if Linearly Independent
- Example #3 – calculate the Wronskian for a 2rd-Order DE and determine if Linearly Independent
- Example #4 – calculate the Wronskian for a 2rd-Order DE and determine if Linearly Independent
- Example #5 – calculate the Wronskian for a 3rd-Order DE and determine if Linearly Independent
- Overview of Able’s Theorem with Example
- Example – calculate the Wronskian using Abel’s Theorem

## Undetermined Coefficients

1 hr 16 min 14 Examples

- Overview of Undetermined Coefficients – Superposition Approach
- More on Undetermined Coefficients and Some Hints and Warnings
- Examples #1-4 – write the form of the Particular Solution to the Inhomogeneous DE
- Examples #5-6 – write the form of the Particular Solution to the Inhomogeneous DE
- Examples #7-8 – write the form of the Particular Solution to the Inhomogeneous DE
- Example #9 – write the form of the Particular Solution to the Inhomogeneous DE
- Example #10 – find the General Solution to the Inhomogeneous DE
- Example #11 – find the General Solution to the Inhomogeneous DE
- Example #12 – find the General Solution to the Inhomogeneous DE
- Example #13 – find the General Solution to the Inhomogeneous DE
- Example #14 – find the General Solution to the Inhomogeneous DE

## Differential Operators

1 hr 48 min 9 Examples

- Quick Review Differential Operators with Examples
- Two Examples – apply the Differential Operator to the indicated function
- Annihilator Method Overview
- Four Examples – find a Differential Operator that Annihilates the given function
- Example – use the Annihilation method to find the General Solution
- Solving Systems by Elimination Overview
- Example #1 – Solve the System by Elimination
- Example #1 – Solve the System by Elimination

## Variation of Parameters

1 hr 9 min 5 Examples

- Overview of Variation of Parameters Method
- Example #1 – find the General Solution to the Inhomogeneous DE with non-constant Coefficients
- Example #2 – find the General Solution to the Inhomogeneous DE with non-constant Coefficients
- Example #3 – find the General Solution to the Inhomogeneous DE
- Example #4 – find the General Solution to the Inhomogeneous DE
- Example #5 – find the General Solution to the Inhomogeneous DE

## Cauchy Euler Equation

23 min 6 Examples

- Overview of the Cauchy-Euler Equation
- Examples #1 & 2 – find the General Solution using the Cauchy-Euler Equation
- Examples #3 & 4 – find the General Solution using the Cauchy-Euler Equation
- Examples #5 & 6 – find the General Solution using the Cauchy-Euler Equation

## Predator Prey Models and Electrical Networks

34 min 4 Examples

- Overview of Predator-Prey Competition Model
- Example #1 – Find the Equilibrium Solution and Population Curve for the Competition Model
- Example #2 – Analyze the Populations over a long pPeriod of time
- Overview of Electrical Networks
- Example #3 – Find the current for the given circuit
- Example #4 – Determine the system of first-order differential equations for the given circuit

## Mechanical Vibrations

1 hr 15 min 6 Examples

- Overview of Mechanical Vibrations
- Steps for Solving Free-Undamped Vibrations
- Example #1 – find the displacement for the Free-Undamped system
- Steps for Solving Free-Damped Vibrations
- Example #2 – find the displacement for the Free-Damped system: Critical Damping
- Example #3 – find the displacement for the Free-Damped system: Over Damping
- Example #4 – find the displacement for the Free-Damped system: Under Damping
- Steps for Solving Forced Vibrations
- Example #5 – find the general solution for the Forced-Undamped System without Resonance
- Example #6 – find the general solution for the Forced-Damped System with Resonance