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.

Preliminaries

1 topics

1.1. Preliminaries

1.1.1.

Differentiating Under the Integral Sign

First-Order Differential Equations

33 topics

2.2. Techniques for Solving First-Order ODEs

	2.2.1.	Solving First-Order ODEs Using Separation of Variables
	2.2.2.	Solving First-Order IVPs Using Separation of Variables
	2.2.3.	Introduction to First-Order Linear ODEs
	2.2.4.	First-Order Linear ODEs With Polynomial Forcing
	2.2.5.	First-Order Linear ODEs With Exponential Forcing
	2.2.6.	First-Order Linear ODEs With Sinusoidal Forcing
	2.2.7.	Solving First-Order ODEs Using Variation of Parameters
	2.2.8.	Solving First-Order ODEs Using Integrating Factors
	2.2.9.	Solving First-Order ODEs by Substitution
	2.2.10.	Further Solving First-Order ODEs by Substitution
	2.2.11.	Reducing ODEs to First-Order Linear by Substitution

2.3. Special First-Order Equations

	2.3.1.	Homogeneous Functions
	2.3.2.	Homogeneous First-Order ODEs
	2.3.3.	Exact Differential Equations
	2.3.4.	Solving Exact ODEs Using Integrating Factors
	2.3.5.	Bernoulli Differential Equations
	2.3.6.	Riccati Differential Equations
	2.3.7.	Clairaut Differential Equations
	2.3.8.	d'Alembert Differential Equations

2.4. Existence, Uniqueness, and Intervals of Validity

	2.4.1.	Intervals of Validity of Differential Equations
	2.4.2.	Existence of Solutions to Differential Equations
	2.4.3.	Uniqueness of Solutions to Differential Equations

2.5. Slope Fields

	2.5.1.	Slope Fields for Directly Integrable Differential Equations
	2.5.2.	Slope Fields for Autonomous Differential Equations
	2.5.3.	Slope Fields for Nonautonomous Differential Equations
	2.5.4.	Analyzing Slope Fields for Directly Integrable Differential Equations
	2.5.5.	Analyzing Slope Fields for Autonomous Differential Equations
	2.5.6.	Analyzing Slope Fields for Nonautonomous Differential Equations

2.6. Qualitative Techniques for Differential Equations

	2.6.1.	Qualitative Analysis of First-Order ODEs
	2.6.2.	Equilibrium Solutions of First-Order ODEs
	2.6.3.	Phase Lines of First-Order ODEs
	2.6.4.	Classifying Equilibrium Solutions of First-Order ODEs
	2.6.5.	Linear Stability Analysis

Modeling With First-Order Differential Equations

22 topics

3.7. Applications of First-Order ODEs

	3.7.1.	Modeling With First-Order ODEs
	3.7.2.	Further Modeling With First-Order ODEs
	3.7.3.	Modeling Mixture Problems With First-Order Separable ODEs
	3.7.4.	Modeling Mixture Problems With First-Order Linear ODEs
	3.7.5.	Orthogonal Trajectories

3.8. Growth and Decay Models With First-Order ODEs

	3.8.1.	Exponential Growth and Decay Models With First-Order ODEs
	3.8.2.	Exponential Growth and Decay Models With First-Order ODEs: Calculating Unknown Times and Initial Values
	3.8.3.	Exponential Growth and Decay Models With First-Order ODEs: Half-Life Problems
	3.8.4.	Inhibited Growth Models With First-Order ODEs
	3.8.5.	Inhibited Decay Models With First-Order ODEs
	3.8.6.	Logistic Growth Models With First-Order ODEs
	3.8.7.	Qualitative Analysis of the Logistic Growth Equation
	3.8.8.	Solving the Logistic Growth Equation

3.9. Physical Applications of First-Order ODEs

	3.9.1.	Velocity and Acceleration as Functions of Displacement
	3.9.2.	Determining Properties of Objects Described as Functions of Displacement
	3.9.3.	Falling Body Problems With Linear Drag
	3.9.4.	Falling Body Problems With Quadratic Drag
	3.9.5.	Newton's Law of Universal Gravitation
	3.9.6.	Modeling Escape Velocity With First-Order ODEs
	3.9.7.	Modeling RL Circuits With First-Order ODEs
	3.9.8.	Modeling RC Circuits With First-Order ODEs
	3.9.9.	Steady-State Solutions of First-Order ODEs

Second-Order Differential Equations

26 topics

4.10. Introduction to Second-Order Linear ODEs

	4.10.1.	Differential Operators
	4.10.2.	Linear Differential Operators
	4.10.3.	Introduction to Second-Order Linear ODEs
	4.10.4.	The Superposition Principle
	4.10.5.	Reduction of Order
	4.10.6.	The Wronskian
	4.10.7.	Abel's Identity
	4.10.8.	General Solutions of Homogeneous Linear ODEs
	4.10.9.	Uniqueness of Solutions for Second-Order Linear ODEs

4.11. Second-Order Linear ODEs With Constant Coefficients

	4.11.1.	Second-Order Homogeneous ODEs: Characteristic Equations With Distinct Real Roots
	4.11.2.	Second-Order Homogeneous ODEs: Characteristic Equations With Repeated Roots
	4.11.3.	Second-Order Homogeneous ODEs: Characteristic Equations With Complex Roots
	4.11.4.	Second-Order Homogeneous ODEs: Initial Value Problems
	4.11.5.	Second-Order Inhomogeneous ODEs With Polynomial Forcing
	4.11.6.	Second-Order Inhomogeneous ODEs With Exponential Forcing
	4.11.7.	Second-Order Inhomogeneous ODEs With Sinusoidal Forcing

4.12. Second-Order Linear ODEs With Variable Coefficients

	4.12.1.	Variation of Parameters for Second-Order ODEs
	4.12.2.	Solving Second-Order ODEs Using Variation of Parameters
	4.12.3.	The Cauchy-Euler Equation: Characteristic Equations With Distinct Real Roots
	4.12.4.	The Cauchy-Euler Equation: Characteristic Equations With Repeated Roots
	4.12.5.	The Cauchy-Euler Equation: Characteristic Equations With Complex Roots
	4.12.6.	The Cauchy-Euler Equation With Forcing

4.13. Higher-Order Linear ODEs

	4.13.1.	Introduction to Nth-Order Linear ODEs
	4.13.2.	Nth-Order Linear Homogeneous Differential Equations
	4.13.3.	Nth-Order Linear Inhomogeneous Differential Equations
	4.13.4.	Variation of Parameters With Nth-Order ODEs

Modeling With Second-Order ODEs

8 topics

5.14. Mechanical Vibrations

	5.14.1.	Simple Harmonic Oscillators
	5.14.2.	Damped Oscillators
	5.14.3.	Forced Oscillators
	5.14.4.	Steady-State Behavior for Vibrating Systems
	5.14.5.	Resonance in Vibrating Systems

5.15. Further Applications of Second-Order ODEs

	5.15.1.	Electrical Circuit Problems
	5.15.2.	Buoyancy Problems
	5.15.3.	Classifying Solutions

Systems of Differential Equations

27 topics

6.16. Systems of ODEs

	6.16.1.	Introduction to Systems of Linear Differential Equations
	6.16.2.	Expressing Homogeneous ODEs as First-Order Systems
	6.16.3.	Expressing Inhomogeneous ODEs as First-Order Systems
	6.16.4.	The Linearity Principle for Systems of Linear ODEs
	6.16.5.	Linear Independence for Systems of Linear ODEs

6.17. Solving Systems of Linear ODEs

	6.17.1.	Solving Decoupled Systems of Linear ODEs
	6.17.2.	Solving Systems of Linear ODEs With Real Distinct Eigenvalues
	6.17.3.	Systems of Linear ODEs: Initial Value Problems
	6.17.4.	Solving Systems of Linear ODEs With Repeated Eigenvalues
	6.17.5.	Solving Systems of Linear ODEs With Complex Eigenvalues
	6.17.6.	Solving Inhomogeneous Systems of Linear ODEs

6.18. Phase Portraits for Systems of Linear ODEs

	6.18.1.	Phase Planes and Phase Portraits
	6.18.2.	Stability of Equilibrium Points for Systems of ODEs
	6.18.3.	Phase Portraits for Decoupled Linear Systems
	6.18.4.	Phase Portraits for Linear Systems With Real Distinct Eigenvalues
	6.18.5.	Phase Portraits for Linear Systems With Repeated Eigenvalues
	6.18.6.	Phase Portraits for Linear Systems With Zero Eigenvalues
	6.18.7.	Phase Portraits for Linear Systems With Complex Eigenvalues
	6.18.8.	Shifted Systems of ODEs
	6.18.9.	Linear Approximations Near Equilibria

6.19. Solving Systems of ODEs Using Matrix Methods

	6.19.1.	Matrix Exponentials
	6.19.2.	Fundamental Matrices
	6.19.3.	Solving Homogeneous Systems of ODEs Using Matrix Methods
	6.19.4.	Solving Inhomogeneous Systems of ODEs Using Matrix Methods
	6.19.5.	Solving Systems of ODEs Using Variation of Parameters

6.20. Modeling With Systems of Linear ODEs

	6.20.1.	The Lotka-Volterra Predator-Prey Model
	6.20.2.	The Lotka-Volterra Model With Carrying Capacity

Laplace Transforms

24 topics

7.21. Laplace Transforms

	7.21.1.	The Unit Step Function
	7.21.2.	Laplace Transforms
	7.21.3.	Linearity of Laplace Transforms
	7.21.4.	Laplace Transforms of Piecewise Functions
	7.21.5.	The First Shifting Theorem
	7.21.6.	The Second Shifting Theorem
	7.21.7.	The Smoothness Property
	7.21.8.	Laplace Transforms of Integrals
	7.21.9.	Existence and Uniqueness of Laplace Transforms

7.22. Inverse Laplace Transforms

	7.22.1.	Inverse Laplace Transforms
	7.22.2.	The First Shifting Theorem for Inverse Laplace Transforms
	7.22.3.	The Second Shifting Theorem for Inverse Laplace Transforms

7.23. Solving Linear ODEs Using Laplace Transforms

	7.23.1.	Laplace Transforms of First Derivatives
	7.23.2.	Laplace Transforms of Second Derivatives
	7.23.3.	Solving First-Order ODEs Using Laplace Transforms
	7.23.4.	Solving Second-Order ODEs Using Laplace Transforms
	7.23.5.	Solving Nth-Order ODEs Using Laplace Transforms
	7.23.6.	Solving Homogeneous Systems of ODEs Using Laplace Transforms
	7.23.7.	Solving Inhomogeneous Systems of ODEs Using Laplace Transforms

7.24. Impulse Forcing

	7.24.1.	The Dirac Delta Function
	7.24.2.	The Laplace Transform of the Dirac Delta Function
	7.24.3.	Solving ODEs With Delta Forcing Using Laplace Transforms
	7.24.4.	Convolutions
	7.24.5.	Convolutions and Delta Forcing

Boundary Value Problems

16 topics

8.25. Fourier Series

	8.25.1.	Introduction to Fourier Series
	8.25.2.	Properties of Fourier Series
	8.25.3.	Fourier Sine Series
	8.25.4.	Fourier Cosine Series
	8.25.5.	Fourier Series of Arbitrary Period
	8.25.6.	Differentiating Fourier Series
	8.25.7.	Integrating Fourier Series
	8.25.8.	Solving ODEs Using Fourier Series
	8.25.9.	Convergence of Fourier Series
	8.25.10.	The Fourier Transform

8.26. Sturm–Liouville Theory

	8.26.1.	Introduction to Boundary Value Problems
	8.26.2.	Second-Order Homogeneous Boundary Value Problems
	8.26.3.	Second-Order Inhomogeneous Boundary Value Problems
	8.26.4.	Eigenvalues and Eigenfunctions of Homogeneous BVPs
	8.26.5.	Regular Sturm–Liouville Systems
	8.26.6.	Properties of Sturm-Liouville Systems

Approximating Solutions to Differential Equations

25 topics

9.27. Series Solutions of Differential Equations

	9.27.1.	Introduction to Recurrence Relations
	9.27.2.	Taylor Series Solutions of Differential Equations
	9.27.3.	Power Series Solutions of Differential Equations
	9.27.4.	Solving Euler's Equation Using Series
	9.27.5.	Regular Singular Points
	9.27.6.	The Method of Frobenius

9.28. Euler's Method

	9.28.1.	Euler's Method: Calculating One Step
	9.28.2.	Euler's Method: Calculating Multiple Steps
	9.28.3.	The Modified Euler Method
	9.28.4.	Euler's Method for Systems of ODEs
	9.28.5.	Euler's Method for Second-Order ODEs

9.29. Higher-Order Numerical Methods

	9.29.1.	The RK4 Method
	9.29.2.	The ABM2 Method
	9.29.3.	The ABM4 and Milne Methods
	9.29.4.	The RK4 Method for Systems of ODEs
	9.29.5.	The RK4 Method for Second-Order ODEs

9.30. Implicit Numerical Methods

	9.30.1.	The Implicit Euler Method
	9.30.2.	The Trapezoidal Method
	9.30.3.	Using the Implicit Euler Method With Newton's Method
	9.30.4.	Using the Trapezoidal Method With Newton's Method

9.31. Analyzing Numerical Methods

	9.31.1.	Big-O Notation
	9.31.2.	Little-O Notation
	9.31.3.	Error in Numerical Methods
	9.31.4.	Stability of Numerical Methods
	9.31.5.	Order of Numerical Methods

Differential Equations

Overview

Outcomes

Content

First-Order Differential Equations

Second-Order Differential Equations

Systems of Linear Differential Equations

Alternative Solution Techniques

Advanced Topics