Math Academy

Upon successful completion of this course, students will have mastered the following:

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.

Advanced Topics

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

37 topics

1.1. First-Order Linear Differential Equations

	1.1.1.	Introduction to First-Order Linear ODEs
	1.1.2.	Solving First-Order Linear ODEs
	1.1.3.	Homogeneous Functions
	1.1.4.	Homogeneous First-Order Differential Equations

1.2. First-Order Nonlinear Differential Equations

	1.2.1.	Solving Differential Equations Using a Linear Substitution
	1.2.2.	Solving Differential Equations Using a Linear Substitution and Factoring
	1.2.3.	Reducing ODEs to First-Order Linear Using a Substitution
	1.2.4.	Exact Differential Equations
	1.2.5.	Solving Exact Differential Equations Using Integrating Factors
	1.2.6.	Bernoulli's Differential Equation
	1.2.7.	Riccati Equations
	1.2.8.	Clairaut Equations

1.3. Existence, Uniqueness, and Intervals of Validity

	1.3.1.	Intervals of Validity of Differential Equations
	1.3.2.	Existence of Solutions to Differential Equations
	1.3.3.	Uniqueness of Solutions to Differential Equations

1.4. Analyzing First-Order Differential Equations

	1.4.1.	The Linearity Principle for First-Order Linear ODEs
	1.4.2.	Steady-State Solutions of First-Order Linear ODEs
	1.4.3.	The Structure Theorem

1.5. Qualitative Techniques for Differential Equations

	1.5.1.	Phase Lines
	1.5.2.	Classifying Equilibrium Solutions
	1.5.3.	Linear Stability Analysis
	1.5.4.	Qualitative Analysis of First-Order Periodic Equations

1.6. Applications of First-Order Differential Equations

	1.6.1.	Modeling With First-Order Linear ODEs
	1.6.2.	Restricted Growth Models With Differential Equations
	1.6.3.	Qualitative Analysis of Restricted Growth and Decay Models
	1.6.4.	Newton's Law of Cooling
	1.6.5.	Modified Logistic Growth Models With Differential Equations
	1.6.6.	Qualitative Analysis of Modified Logistic Growth Models
	1.6.7.	Velocity and Acceleration as Functions of Displacement
	1.6.8.	Determining Properties of Objects Described as Functions of Displacement
	1.6.9.	Falling Body Problems
	1.6.10.	Escape Velocity
	1.6.11.	Planetary Motion
	1.6.12.	Particles Moving Along Curves
	1.6.13.	Dilution Problems
	1.6.14.	Electrical Circuits
	1.6.15.	Orthogonal Trajectories

Second-Order Differential Equations

30 topics

2.7. Introduction to Homogeneous Linear ODEs

	2.7.1.	Linear Differential Operators
	2.7.2.	Introduction to Second-Order Linear ODEs
	2.7.3.	The Superposition Principle
	2.7.4.	Reduction of Order
	2.7.5.	The Wronskian and Linear Independence
	2.7.6.	Abel's Identity
	2.7.7.	General Solutions of Homogeneous Linear ODEs
	2.7.8.	Uniqueness of Solutions for Second-Order Linear ODEs

2.8. Second-Order Homogeneous ODEs with Constant Coefficients

	2.8.1.	Second-Order Homogeneous ODEs: Characteristic Equations With Distinct Real Roots
	2.8.2.	Second-Order Homogeneous ODEs: Characteristic Equations With Repeated Roots
	2.8.3.	Second-Order Homogeneous ODEs: Characteristic Equations With Complex Roots
	2.8.4.	Second-Order Homogeneous ODEs: Initial Value Problems

2.9. Second-Order Inhomogeneous ODEs with Constant Coefficients

	2.9.1.	Second-Order ODEs With Polynomial Forcing
	2.9.2.	Second-Order ODEs With Exponential Forcing
	2.9.3.	Second-Order ODEs With Sinusoidal Forcing
	2.9.4.	The Method of Variation of Parameters

2.10. The Cauchy-Euler Equation

	2.10.1.	The Cauchy-Euler Equation: Characteristic Equations With Distinct Real Roots
	2.10.2.	The Cauchy-Euler Equation: Characteristic Equations With Repeated Roots
	2.10.3.	The Cauchy-Euler Equation: Characteristic Equations With Complex Roots
	2.10.4.	The Cauchy-Euler Equation With Forcing

2.11. Special Second-Order Linear ODEs

	2.11.1.	Airy's Differential Equation
	2.11.2.	Bessel's Differential Equation
	2.11.3.	Chebyshev Differential Equation
	2.11.4.	Hermite Differential Equation
	2.11.5.	Laguerre Differential Equation
	2.11.6.	Legendre Differential Equation

2.12. Higher-Order Linear ODEs

	2.12.1.	Introduction to Nth-Order Linear ODEs
	2.12.2.	Nth-Order Linear Homogeneous Differential Equations
	2.12.3.	Nth-Order Linear Inhomogeneous Differential Equations
	2.12.4.	Variation of Parameters With Nth-Order ODEs

Modeling With Second-Order Differential Equations

8 topics

3.13. Mechanical Vibrations

	3.13.1.	Simple Harmonic Oscillators
	3.13.2.	Damped Oscillators
	3.13.3.	Forced Oscillators
	3.13.4.	Steady-State Behavior for Vibrating Systems
	3.13.5.	Resonance in Vibrating Systems

3.14. Further Applications of Second-Order ODEs

	3.14.1.	Electrical Circuit Problems
	3.14.2.	Buoyancy Problems
	3.14.3.	Classifying Solutions

Laplace Transforms

18 topics

4.15. Laplace Transforms

	4.15.1.	The Dirac Delta Function
	4.15.2.	The Unit Step Function
	4.15.3.	Laplace Transforms
	4.15.4.	Properties of Laplace Transforms
	4.15.5.	Calculating Laplace Transforms Using Tables
	4.15.6.	Inverse Laplace Transforms
	4.15.7.	The First Shifting Theorem
	4.15.8.	The Second Shifting Theorem
	4.15.9.	Existence of Laplace Transforms
	4.15.10.	Laplace Transforms of Derivatives
	4.15.11.	Laplace Transforms of Integrals
	4.15.12.	The Laplace Transform of the Dirac-Delta Function
	4.15.13.	Convolutions

4.16. Solving Linear ODEs Using Laplace Transforms

	4.16.1.	Solving First-Order ODEs Using Laplace Transforms
	4.16.2.	Solving Second-Order ODEs Using Laplace Transforms
	4.16.3.	Solving Nth-Order ODEs Using Laplace Transforms
	4.16.4.	Solving ODEs With Delta Forcing Using Laplace Transforms
	4.16.5.	Convolutions and Delta Forcing

Systems of Differential Equations

29 topics

5.17. Systems of Linear Differential Equations

	5.17.1.	Introduction to Systems of Linear Differential Equations
	5.17.2.	Expressing Second-Order and Third-Order Homogeneous ODEs as First-Order Systems
	5.17.3.	Expressing Second-Order and Third-Order Inhomogeneous ODEs as First-Order Systems
	5.17.4.	The Linearity Principle for Systems of Linear ODEs

5.18. Homogeneous Systems of Linear ODEs

	5.18.1.	Solving Decoupled Systems of Linear ODEs
	5.18.2.	Solving Systems of Linear ODEs With Real Distinct Eigenvalues
	5.18.3.	Systems of Linear ODEs: Initial Value Problems
	5.18.4.	Solving Systems of Linear ODEs With Repeated Eigenvalues
	5.18.5.	Solving Systems of Linear ODEs With Complex Eigenvalues
	5.18.6.	Solving Homogeneous Systems of ODEs Using Laplace Transforms

5.19. Fundamental Matrix

	5.19.1.	Fundamental Matrix
	5.19.2.	Matrix Exponential
	5.19.3.	Linear Independence for Systems of Linear ODEs

5.20. Inhomogeneous Systems of Linear ODEs

	5.20.1.	Introduction to Inhomogeneous Systems of Linear ODEs
	5.20.2.	Solving Inhomogeneous Systems of Linear ODEs Using Variation of Parameters
	5.20.3.	Solving Inhomogeneous Systems of Linear ODEs Using Laplace Transforms

5.21. Phase Portraits for Systems of Linear ODEs

	5.21.1.	Phase Planes and Phase Portraits
	5.21.2.	Stability, Asymptotic Stability, and Instability
	5.21.3.	Linear Approximations Near Equilibria
	5.21.4.	Phase Portraits for Decoupled Linear Systems
	5.21.5.	Phase Portraits for Linear Systems With Real Distinct Eigenvalues
	5.21.6.	Phase Portraits for Linear Systems With Repeated Eigenvalues
	5.21.7.	Phase Portraits for Linear Systems With Zero Eigenvalues
	5.21.8.	Phase Portraits for Linear Systems With Complex Eigenvalues
	5.21.9.	Shifted Systems

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

Boundary Value Problems

23 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
	6.23.5.	Green's Function

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.	Fourier Series of Arbitrary Period
	6.24.5.	Differentiating Fourier Series
	6.24.6.	Integrating Fourier Series
	6.24.7.	Solving ODEs Using Fourier Series
	6.24.8.	Convergence of Fourier Series
	6.24.9.	The Fourier Transform

6.25. Sturm-Liouville Theory

	6.25.1.	The Regular Sturm-Liouville Problem
	6.25.2.	Properties of Sturm-Liouville Eigenvalue Problems
	6.25.3.	The Sturm Comparison Theorem
	6.25.4.	Lagrange's Identity
	6.25.5.	Green's Identity
	6.25.6.	Orthogonality of Eigenfunctions
	6.25.7.	Reality of Eigenvalues
	6.25.8.	The Rayleigh Quotient
	6.25.9.	Eigenfunction Expansion

Series and Numerical Solutions of Differential Equations

16 topics

7.26. Series Solutions of Differential Equations

	7.26.1.	Taylor Series Solutions of Differential Equations
	7.26.2.	Power Series Solutions of Differential Equations
	7.26.3.	Solving Euler's Equation Using Series
	7.26.4.	Regular Singular Points
	7.26.5.	The Method of Frobenius

7.27. Numerical Solutions of Differential Equations

	7.27.1.	Error and Stability in Euler's Method
	7.27.2.	The Modified Euler Method for First-Order ODEs
	7.27.3.	The Implicit Euler Method
	7.27.4.	The Runge-Kutta Method for First-Order Equations
	7.27.5.	The Adams-Bashford-Moulton Method for First-Order ODEs
	7.27.6.	Milne's Method for First-Order ODEs
	7.27.7.	The Order of a Numerical Method
	7.27.8.	Euler's Method for Systems of ODEs
	7.27.9.	Euler's Method for Second-Order ODEs
	7.27.10.	The Runge-Kutta Method for Second-Order Equations
	7.27.11.	The Adams-Bashford-Moulton for Second-Order ODEs

Differential Equations

Overview

Outcomes

Content

First-Order Differential Equations

Second-Order Differential Equations

Systems of Linear Differential Equations

Alternative Solution Techniques

Advanced Topics