This course is currently under construction.
The target release date for this course is **July, 2024**.

Master a variety of techniques for solving equations that arise when using calculus to model real-world situations.

- Solve first-order differential equations using integrating factors.
- Identify and apply substitutions to convert first-order differential equations into forms that permit known solution techniques including separation of variables and integrating factors.
- Solve special first-order differential equations including exact equations and instances of Bernoulli’s equation.
- Reason about the solutions of differential equations using the existence and uniqueness theorem.
- Leverage qualitative techniques such as phase lines, equilibrium solutions, and bifurcation diagrams to infer properties of solutions of first-order differential equations.
- Construct and solve first-order differential equation models in a variety of real-world modeling contexts including population growth, temperature cooling, mechanics, and more.

- Use reduction of order to reduce second-order differential equations to first-order differential equations for which solution techniques are known.
- Use the characteristic equation in conjunction with the superposition principle to solve homogeneous linear second-order differential equations.
- Interpret the right-hand side of an inhomogeneous differential equation as a forcing function, and use the method of undetermined coefficients to find particular solutions for various types of forcing functions.
- Use variation of parameters as a general technique for finding a particular solution to an inhomogeneous second-order differential equation.
- Solve instances of the Cauchy-Euler equation.
- Construct and solve second-order differential equation models in a variety of real-world modeling contexts including oscillators and vibrating systems.
- Extend second-order solution techniques to Nth-order linear differential equations.

- Solve homogeneous systems of linear differential equations by computing eigenvalues and eigenvectors.
- Express higher-order differential equations as first-order systems.
- Extend qualitative techniques and variation of parameters to systems of linear differential equations.
- Construct and solve systems of differential equations in a variety of real-world modeling contexts including predator-prey populations and the motion of a mass on a spring.

- Compute Laplace transforms and use them to solve differential equations.
- Find the recurrence relation that generates a power series solution for a given differential equation.
- Employ numerical techniques including Euler’s method and the Runge-Kutta method to estimate solutions to initial value problems.

- Use perturbation theory to find approximate solutions to differential equations starting from exact solutions in simpler cases.
- Construct and solve partial differential equations in a variety of real-world modeling contexts including heat diffusion, waves, and vibrating strings.

1.

First-Order Differential Equations
31 topics

1.1. First-Order Linear Differential Equations

1.1.1. | Introduction to First-Order Linear Differential Equations | |

1.1.2. | The Linearity Principle for First-Order Linear Equations | |

1.1.3. | Steady-State Solutions | |

1.1.4. | Solving First-Order Linear ODEs Using Integrating Factors | |

1.1.5. | Modeling With First-Order Linear Differential Equations |

1.2. Solving Differential Equations by Substitution

1.2.1. | Homogeneous Functions | |

1.2.2. | Homogeneous First-Order Differential Equations | |

1.2.3. | Solving Differential Equations Using a Linear Substitution | |

1.2.4. | Solving Differential Equations Using a Linear Substitution and Factoring | |

1.2.5. | Reducing ODEs to First-Order Linear Using a Substitution |

1.3. Special First-Order Differential Equations

1.3.1. | Exact Differential Equations | |

1.3.2. | Bernoulli's Differential Equation |

1.4. Existence, Uniqueness, and Intervals of Validity

1.4.1. | Intervals of Validity of Differential Equations | |

1.4.2. | Existence of Solutions to Differential Equations | |

1.4.3. | Uniqueness of Solutions to Differential Equations |

1.5. Qualitative Techniques for Differential Equations

1.5.1. | Phase Lines | |

1.5.2. | Classifying Equilibrium Solutions |

1.6. Introduction to Bifurcation Theory

1.6.1. | One Parameter Families of Solutions | |

1.6.2. | Bifurcation Diagrams | |

1.6.3. | Bifurcations of Equilibrium Points |

1.7. Applications of First-Order Differential Equations

1.7.1. | Restricted Growth Models With Differential Equations | |

1.7.2. | Qualitative Analysis of Restricted Growth and Decay Models | |

1.7.3. | Newton's Law of Cooling | |

1.7.4. | Modified Logistic Growth Models With Differential Equations | |

1.7.5. | Qualitative Analysis of Modified Logistic Growth Models | |

1.7.6. | Velocity and Acceleration as Functions of Displacement | |

1.7.7. | Determining Properties of Objects Described as Functions of Displacement | |

1.7.8. | Falling Body Problems | |

1.7.9. | Dilution Problems | |

1.7.10. | Electrical Circuits | |

1.7.11. | Orthogonal Trajectories |

2.

Second-Order Differential Equations
28 topics

2.8. Introduction to Homogeneous Linear ODEs

2.8.1. | Linear Differential Operators | |

2.8.2. | Introduction to Linear Differential Equations | |

2.8.3. | The Superposition Principle | |

2.8.4. | Reduction of Order | |

2.8.5. | The Wronskian and Linear Independence | |

2.8.6. | General Solutions of Homogeneous Linear ODEs |

2.9. Second-Order Homogeneous ODEs with Constant Coefficients

2.9.1. | Second-Order Homogeneous ODEs: Characteristic Equations With Distinct Real Roots | |

2.9.2. | Second-Order Homogeneous ODEs: Characteristic Equations With Repeated Roots | |

2.9.3. | Second-Order Homogeneous ODEs: Characteristic Equations With Complex Roots | |

2.9.4. | Second-Order Homogeneous ODEs: Initial Value Problems |

2.10. Second-Order Inhomogeneous ODEs with Constant Coefficients

2.10.1. | Second-Order ODEs With Polynomial Forcing | |

2.10.2. | Second-Order ODEs With Exponential Forcing | |

2.10.3. | Second-Order ODEs With Sinusoidal Forcing | |

2.10.4. | The Method of Variation of Parameters |

2.11. The Cauchy-Euler Equation

2.11.1. | The Cauchy-Euler Equation: Characteristic Equations With Distinct Real Roots | |

2.11.2. | The Cauchy-Euler Equation: Characteristic Equations With Repeated Roots | |

2.11.3. | The Cauchy-Euler Equation: Characteristic Equations With Complex Roots | |

2.11.4. | The Cauchy-Euler Equation With Forcing |

2.12. Special Second-Order Linear ODEs

2.12.1. | Airy's Differential Equation | |

2.12.2. | Bessel's Differential Equation | |

2.12.3. | Chebyshev Differential Equation | |

2.12.4. | Hermite Differential Equation | |

2.12.5. | Laguerre Differential Equation | |

2.12.6. | Legendre Differential Equation |

2.13. Higher-Order Linear ODEs

2.13.1. | Introduction to Nth-Order Linear ODEs | |

2.13.2. | Nth-Order Linear Homogeneous Differential Equations | |

2.13.3. | Nth-Order Linear Inhomogeneous Differential Equations | |

2.13.4. | Variation of Parameters With Nth-Order ODEs |

3.

Modeling With Second-Order Differential Equations
8 topics

3.14. Mechanical Vibrations

3.14.1. | Simple Harmonic Oscillators | |

3.14.2. | Damped Oscillators | |

3.14.3. | Forced Oscillators | |

3.14.4. | Steady-State Behavior for Vibrating Systems | |

3.14.5. | Resonance in Vibrating Systems |

3.15. Further Applications of Second-Order ODEs

3.15.1. | Electrical Circuit Problems | |

3.15.2. | Buoyancy Problems | |

3.15.3. | Classifying Solutions |

4.

Laplace Transforms
12 topics

4.16. Laplace Transforms

4.16.1. | The Dirac Delta Function | |

4.16.2. | The Unit Step Function | |

4.16.3. | Laplace Transforms | |

4.16.4. | Calculating Laplace Transforms Using Tables | |

4.16.5. | Laplace Transforms of Derivatives | |

4.16.6. | Inverse Laplace Transforms | |

4.16.7. | Convolutions |

4.17. Solving Linear ODEs Using Laplace Transforms

4.17.1. | Solving First-Order ODEs Using Laplace Transforms | |

4.17.2. | Solving Second-Order ODEs Using Laplace Transforms | |

4.17.3. | Solving Nth-Order ODEs Using Laplace Transforms | |

4.17.4. | Solving ODEs With Delta Forcing Using Laplace Transforms | |

4.17.5. | Convolutions and Delta Forcing |

5.

Systems of Differential Equations
22 topics

5.18. Systems of Linear Differential Equations

5.18.1. | Introduction to Systems of Linear Differential Equations | |

5.18.2. | Expressing Second-Order and Third-Order Homogeneous ODEs as First-Order Systems | |

5.18.3. | Expressing Second-Order and Third-Order Inhomogeneous ODEs as First-Order Systems | |

5.18.4. | Phase Planes and Phase Portraits | |

5.18.5. | The Linearity Principle for Systems of Linear ODEs |

5.19. Homogeneous Systems of Linear ODEs

5.19.1. | Solving Decoupled Systems of Linear ODEs | |

5.19.2. | Solving Systems of Linear ODEs With Real Distinct Eigenvalues | |

5.19.3. | Systems of Linear ODEs: Initial Value Problems | |

5.19.4. | Solving Systems of Linear ODEs With Repeated Eigenvalues | |

5.19.5. | Solving Systems of Linear ODEs With Complex Eigenvalues | |

5.19.6. | Solving Homogeneous Systems of ODEs Using Laplace Transforms |

5.20. Phase Portraits for Systems of Linear ODEs

5.20.1. | Phase Portraits for Decoupled Linear Systems | |

5.20.2. | Phase Portraits for Linear Systems With Real Distinct Eigenvalues | |

5.20.3. | Phase Portraits for Linear Systems With Repeated Eigenvalues | |

5.20.4. | Phase Portraits for Linear Systems With Complex Eigenvalues |

5.21. Inhomogeneous Systems of Linear ODEs

5.21.1. | Introduction to Inhomogeneous Systems of Linear ODEs | |

5.21.2. | Solving Inhomogeneous Systems of Linear ODEs Using Variation of Parameters | |

5.21.3. | Solving Inhomogeneous Systems of Linear ODEs Using Laplace Transforms |

5.22. Modeling With Systems of Linear ODEs

5.22.1. | The Predator-Prey Model | |

5.22.2. | The Revised Predator-Prey Model | |

5.22.3. | Modeling Mass-Spring Systems | |

5.22.4. | The Lorentz Equations |

6.

Boundary Value Problems
8 topics

6.23. Introduction to Boundary Value Problems

6.23.1. | Second-Order Homogeneous ODEs: Boundary Value Problems | |

6.23.2. | Classification of Boundary Conditions | |

6.23.3. | Eigenvalues and Eigenfunctions for Boundary Value Problems | |

6.23.4. | Orthogonal Functions |

6.24. Fourier Series

6.24.1. | Introduction to Fourier Series | |

6.24.2. | Fourier Sine Series | |

6.24.3. | Fourier Cosine Series | |

6.24.4. | Convergence of Fourier Series |

7.

Series and Numerical Solutions of Differential Equations
10 topics

7.25. Series Solutions of Differential Equations

7.25.1. | Taylor Series Solutions of Differential Equations | |

7.25.2. | Power Series Solutions of Differential Equations | |

7.25.3. | Solving Euler's Equation Using Series | |

7.25.4. | Regular Singular Points | |

7.25.5. | The Method of Frobenius |

7.26. Numerical Solutions of Differential Equations

7.26.1. | Error and Stability in Euler's Method | |

7.26.2. | The Modified Euler Method | |

7.26.3. | Euler's Method for Systems of ODEs | |

7.26.4. | The Runge-Kutta Method for First-Order Equations | |

7.26.5. | The Runge-Kutta Method for Second-Order Equations |

8.

Perturbation Methods
6 topics

8.27. Introduction to Perturbation Theory

8.27.1. | Introduction to Perturbation Theory | |

8.27.2. | Regular Asymptotic Solutions of Polynomial Equations | |

8.27.3. | Regular Asymptotic Solutions of Differential Equations | |

8.27.4. | Singular Asymptotic Solutions of Polynomial Equations | |

8.27.5. | Singular Asymptotic Solutions of Differential Equations | |

8.27.6. | The Poincaré-Lindstedt Method |

9.

Partial Differential Equations
9 topics

9.28. Partial Differential Equations

9.28.1. | Introduction to Partial Differential Equations | |

9.28.2. | The Laplacian Operator in Cartesian Coordinates | |

9.28.3. | The Heat Equation | |

9.28.4. | The Wave Equation | |

9.28.5. | Separation of Variables for Partial Differential Equations | |

9.28.6. | Solving the Heat Equation | |

9.28.7. | Solving the Heat Equation With Non-Zero Temperature Boundaries | |

9.28.8. | Laplace's Equation | |

9.28.9. | The Vibrating String Problem |