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.
First-Order Differential Equations
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.
Second-Order Differential Equations
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.
Systems of 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.
Alternative Solution Techniques
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.
First-Order Differential Equations
1.1. First-Order Linear Differential Equations
Introduction to First-Order Linear Differential Equations
The Linearity Principle for First-Order Linear Equations
Solving First-Order Linear ODEs Using Integrating Factors
Modeling With First-Order Linear Differential Equations
1.2. Solving Differential Equations by Substitution
Homogeneous First-Order Differential Equations
Solving Differential Equations Using a Linear Substitution
Solving Differential Equations Using a Linear Substitution and Factoring
Reducing ODEs to First-Order Linear Using a Substitution
1.3. Special First-Order Differential Equations
Exact Differential Equations
Bernoulli's Differential Equation
1.4. Existence, Uniqueness, and Intervals of Validity
Intervals of Validity of Differential Equations
Existence of Solutions to Differential Equations
Uniqueness of Solutions to Differential Equations
1.5. Qualitative Techniques for Differential Equations
Classifying Equilibrium Solutions
1.6. Introduction to Bifurcation Theory
One Parameter Families of Solutions
Bifurcations of Equilibrium Points
1.7. Applications of First-Order Differential Equations
Restricted Growth Models With Differential Equations
Qualitative Analysis of Restricted Growth and Decay Models
Newton's Law of Cooling
Modified Logistic Growth Models With Differential Equations
Qualitative Analysis of Modified Logistic Growth Models
Velocity and Acceleration as Functions of Displacement
Determining Properties of Objects Described as Functions of Displacement
Falling Body Problems
Second-Order Differential Equations
2.8. Introduction to Homogeneous Linear ODEs
Linear Differential Operators
Introduction to Linear Differential Equations
The Superposition Principle
Reduction of Order
The Wronskian and Linear Independence
General Solutions of Homogeneous Linear ODEs
2.9. Second-Order Homogeneous ODEs with Constant Coefficients
Second-Order Homogeneous ODEs: Characteristic Equations With Distinct Real Roots
Second-Order Homogeneous ODEs: Characteristic Equations With Repeated Roots
Second-Order Homogeneous ODEs: Characteristic Equations With Complex Roots
Second-Order Homogeneous ODEs: Initial Value Problems
2.10. Second-Order Inhomogeneous ODEs with Constant Coefficients
Second-Order ODEs With Polynomial Forcing
Second-Order ODEs With Exponential Forcing
Second-Order ODEs With Sinusoidal Forcing
The Method of Variation of Parameters
2.11. The Cauchy-Euler Equation
The Cauchy-Euler Equation: Characteristic Equations With Distinct Real Roots
The Cauchy-Euler Equation: Characteristic Equations With Repeated Roots
The Cauchy-Euler Equation: Characteristic Equations With Complex Roots
The Cauchy-Euler Equation With Forcing
2.12. Special Second-Order Linear ODEs
Airy's Differential Equation
Bessel's Differential Equation
Chebyshev Differential Equation
Hermite Differential Equation
Laguerre Differential Equation
Legendre Differential Equation
2.13. Higher-Order Linear ODEs
Introduction to Nth-Order Linear ODEs
Nth-Order Linear Homogeneous Differential Equations
Nth-Order Linear Inhomogeneous Differential Equations
Variation of Parameters With Nth-Order ODEs
Modeling With Second-Order Differential Equations
3.14. Mechanical Vibrations
Simple Harmonic Oscillators
Steady-State Behavior for Vibrating Systems
Resonance in Vibrating Systems
3.15. Further Applications of Second-Order ODEs
Electrical Circuit Problems
4.16. Laplace Transforms
The Dirac Delta Function
The Unit Step Function
Calculating Laplace Transforms Using Tables
Laplace Transforms of Derivatives
Inverse Laplace Transforms
4.17. Solving Linear ODEs Using Laplace Transforms
Solving First-Order ODEs Using Laplace Transforms
Solving Second-Order ODEs Using Laplace Transforms
Solving Nth-Order ODEs Using Laplace Transforms
Solving ODEs With Delta Forcing Using Laplace Transforms
Convolutions and Delta Forcing
Systems of Differential Equations
5.18. Systems of Linear Differential Equations
Introduction to Systems of Linear Differential Equations
Expressing Second-Order and Third-Order Homogeneous ODEs as First-Order Systems
Expressing Second-Order and Third-Order Inhomogeneous ODEs as First-Order Systems
Phase Planes and Phase Portraits
The Linearity Principle for Systems of Linear ODEs
5.19. Homogeneous Systems of Linear ODEs
Solving Decoupled Systems of Linear ODEs
Solving Systems of Linear ODEs With Real Distinct Eigenvalues
Systems of Linear ODEs: Initial Value Problems
Solving Systems of Linear ODEs With Repeated Eigenvalues
Solving Systems of Linear ODEs With Complex Eigenvalues
Solving Homogeneous Systems of ODEs Using Laplace Transforms
5.20. Phase Portraits for Systems of Linear ODEs
Phase Portraits for Decoupled Linear Systems
Phase Portraits for Linear Systems With Real Distinct Eigenvalues
Phase Portraits for Linear Systems With Repeated Eigenvalues
Phase Portraits for Linear Systems With Complex Eigenvalues
5.21. Inhomogeneous Systems of Linear ODEs
Introduction to Inhomogeneous Systems of Linear ODEs
Solving Inhomogeneous Systems of Linear ODEs Using Variation of Parameters
Solving Inhomogeneous Systems of Linear ODEs Using Laplace Transforms
5.22. Modeling With Systems of Linear ODEs
The Predator-Prey Model
The Revised Predator-Prey Model
Modeling Mass-Spring Systems
The Lorentz Equations
Boundary Value Problems
6.23. Introduction to Boundary Value Problems
Second-Order Homogeneous ODEs: Boundary Value Problems
Classification of Boundary Conditions
Eigenvalues and Eigenfunctions for Boundary Value Problems
6.24. Fourier Series
Introduction to Fourier Series
Fourier Sine Series
Fourier Cosine Series
Convergence of Fourier Series
Series and Numerical Solutions of Differential Equations
7.25. Series Solutions of Differential Equations
Taylor Series Solutions of Differential Equations
Power Series Solutions of Differential Equations
Solving Euler's Equation Using Series
Regular Singular Points
The Method of Frobenius
7.26. Numerical Solutions of Differential Equations
Error and Stability in Euler's Method
The Modified Euler Method
Euler's Method for Systems of ODEs
The Runge-Kutta Method for First-Order Equations
The Runge-Kutta Method for Second-Order Equations
8.27. Introduction to Perturbation Theory
Introduction to Perturbation Theory
Regular Asymptotic Solutions of Polynomial Equations
Regular Asymptotic Solutions of Differential Equations
Singular Asymptotic Solutions of Polynomial Equations
Singular Asymptotic Solutions of Differential Equations
The Poincaré-Lindstedt Method
Partial Differential Equations
9.28. Partial Differential Equations
Introduction to Partial Differential Equations
The Laplacian Operator in Cartesian Coordinates
The Heat Equation
The Wave Equation
Separation of Variables for Partial Differential Equations
Solving the Heat Equation
Solving the Heat Equation With Non-Zero Temperature Boundaries