**First Order Differential Equations**

## Directional Fields

45 min 5 Examples

- Quick Review of Solutions of a Differential Equation and Steps for an IVP
- Example #1 – sketch the direction field by hand
- Example #2 – sketch the direction field for a logistic differential equation
- Isoclines Definition and Example
- Autonomous Differential Equations and Equilibrium Solutions
- Overview of Autonomously Stable, Unstable, and Semistable Equilibrium Solutions
- Steps for finding Autonomously Stable, Unstable, and Semistable Equilibrium Solutions with Example
- Example – Find and Classify all Equilibrium Solutions for the Autonomous DE

## Separable Equations

56 min 7 Examples

- Overview of Separable Differential Equation and Steps for Solving
- Example #1 – find a one-parameter family of solutions
- Example #2 – find a one-parameter family of solutions using Integration by Parts
- Example #3 – find a one-parameter family of solutions using Long-Division
- Example #4 – find a one-parameter family of solutions for the IVP
- Example #5 – find a one-parameter family of solutions for the IVP using the Quadratic Formula
- Example #6 – find a one-parameter family of solutions for the IVP
- Example #7 – find a one-parameter family of solutions for the IVP and find the Singular Solution

## Euler’s Method

35 min 3 Examples

- Overview of Eulee’s Method
- Example #1: Use Euler’s Method to approximate the Initial Value Problem
- Example #2: Use Euler’s Method to approximate the Initial Value Problem
- Example #3: Use Euler’s Method and Discuss the Error in the Euler Approximation

## First Order Linear Differential Equation

1 hr 24 min 6 Examples

- Overview of Linear Differential using an Integrating Factor and Steps for Solving
- Example #1 – solve the Linear First-Order Differential Equation
- Example #2 – solve the Linear First-Order Differential Equation
- Example #3 – solve the Linear First-Order Differential Equation
- Example #4 – solve the Linear First-Order Differential Equation given an Initial Condition
- Example #5 – solve the Linear First-Order Differential Equation given an Initial Condition
- Example #6 – solve the Piecewise Linear First-Order Differential Equation

## Solving Exact ODEs

1 hr 10 min 8 Examples

- Overview and Steps for Identifying and Solving Exact Differential Equations
- Example #1 – find the solution to the Exact Differential Equation
- Example #2 – find the solution to the Exact Differential Equation
- Example #3 – find the solution to the Exact Differential Equation
- Example #4 – solve the Exact Differential Equation given an IVP
- Example #5 – solve the Exact Differential Equation given an IVP
- Example #6 – solve the Exact Differential Equation given an IVP
- Steps for solving a DE by making it an Exact Differential Equation
- Example #7 – solve the given DE by making it Exact
- Example #8 – solve the given DE by making it Exact

## Homogeneous Differential Equation

1 hr 28 min 5 Examples

- Determining if a function is Homogeneous and identify the degree
- Steps for Solving Homogeneous First Order ODEs
- Example #1 – solve the Homogeneous First Order DE
- Example #2 – solve the Homogeneous First Order DE
- Example #3 – solve the Homogeneous First Order DE
- Example #4 – solve the Homogeneous First Order DE given an Initial Condition
- Example #5 – solve the Homogeneous First Order DE given an Initial Condition

## Bernoulli Differential Equation

50 min 4 Examples

- Overview and Steps for Solving a Bernoulli Differential Equation
- Example #1 – solve the Bernoulli DE
- Example #2 – solve the Bernoulli DE
- Example #3 – solve the Bernoulli DE given an Initial Condition
- Example #4 – solve the Bernoulli DE given an Initial Condition

## Linear and Nonlinear Models

1 hr 59 min 9 Examples

- Overview of Population Dynamics
- Example #1 – population of a community
- Example #2 – radioactive decay
- Example #3 – spread of disease using the logistic DE
- Example #4 – compound interest
- Overview of Mixtures
- Example #5 – well mixed solution where input and output rates are the same
- Example #6 – well mixed solution where input and output rates are different
- Overview of Falling Bodies and Air Resistance
- Example #7 – determine equation of motion and when an object hits the ground
- Overview of Newton’s Law of Cooling
- Example #8 – when will the food reach a certain temperature?
- Example #9 – determine the time of death