Direction 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
- Discover a one-parameter solution family (Example #1)
- Utilize Integration by Parts to obtain a solution family (Example #2)
- Employ Long-Division for a solution family (Example #3)
- Determine a solution family for the IVP (Example #4)
- Apply Quadratic Formula for IVP solution family (Example #5)
- Acquire a one-parameter family for the IVP (Example #6)
- Obtain IVP solution family and Singular Solution (Example #7)
Euler’s Method
35 min 3 Examples
- An overview of Euler’s Method
- Approximate the Initial Value Problem using Euler’s Method (Example #1)
- Initial Value Problem approximation with Euler’s Method (Example #2)
- Employ Euler’s Method, discussing the error in its approximation (Example #3)
First Order Linear Differential Equation
1 hr 24 min 6 Examples
- Integrating Factor and Steps for Solving Linear Differential Equations: An Overview
- Solving Linear First-Order Differential Equations (Examples #1-3)
- Linear First-Order Differential Equations with Initial Conditions (Examples #4-5)
- Piecewise Linear First-Order Differential Equation (Example #6)
Solving Exact ODEs
1 hr 10 min 8 Examples
- Overview and Steps for Identifying and Solving Exact Differential Equations
- Find the solution to the Exact Differential Equation (Examples #1, #2, and #3)
- Solve the Exact Differential Equation given an IVP (Examples #4, #5, and #6)
- Steps for solving a DE by making it an Exact Differential Equation and solving examples (Examples #7 and #8)
Homogeneous Differential Equation
1 hr 28 min 5 Examples
- Determining if a function is Homogeneous and identifying the degree
- Steps for Solving Homogeneous First Order ODEs
- Solving Homogeneous First Order DEs: Examples (Examples #1-3)
- Solving Homogeneous First Order DEs given an Initial Condition: Examples (Examples #4-5)
Bernoulli Differential Equation
50 min 4 Examples
- Overview and Steps for Solving a Bernoulli Differential Equation
- Solving Bernoulli DEs: Examples (Examples #1-2)
- Solving Bernoulli DEs given an Initial Condition: Examples (Examples #3-4)
Linear and Nonlinear Models
1 hr 59 min 9 Examples
- Overview of Population Dynamics
- Population Dynamics Examples (Examples #1-4): community population, radioactive decay, spread of disease using the logistic DE, compound interest
- Overview of Mixtures
- Mixtures Examples (Examples #5-6): well mixed solution with equal input and output rates, well mixed solution with different input and output rates
- Overview of Falling Bodies and Air Resistance
- Falling Bodies and Air Resistance Example (Example #7): determining equation of motion and when an object hits the ground
- Overview of Newton’s Law of Cooling
- Newton’s Law of Cooling Examples (Examples #8-9): determining when food will reach a certain temperature, determining the time of death
Predator Prey Models and Electrical Networks
34 min 4 Examples
- Overview of Predator-Prey Competition Model
- Find Equilibrium Solution & Population Curve (Example #1)
- Analyze Populations over Long Time Periods (Example #2)
- Overview of Electrical Networks
- Find Current for the Circuit (Example #3)
- Determine First-Order DE System for Circuit (Example #4)
Chapter Test
1 hr 50 min 10 Problems
- Draw direction field and obtain particular solution (Problem #1a-b)
- Create phase portrait for autonomous DE and categorize critical points (Problem #2)
- Approximate with Euler’s method (Problem #3)
- Determine explicit solution via separation of variables (Problem #4)
- Obtain general solution employing an integrating factor (Problem #5)
- Resolve IVP, providing largest interval for solution definition (Problem #6)
- Address IVP using integrating factor to achieve Exact equation (Problem #7)
- Apply suitable substitution to solve DE (Problem #8)
- Resolve Bernoulli equation (Problem #9)
- Implement Newton’s Law of Cooling (Problem #10)