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Differential Equations and Their Solutions

A differential equation is an equation for an unknown function that involves its derivatives, and this unit builds the language the rest of the course speaks. Topics: Classifying Differential Equations, Solutions and Solution Families, Initial-Value Problems, Existence and Uniqueness, Direction Fields, Autonomous Equations and the Phase Line.

First-Order Methods

This unit is the toolbox for solving first-order equations exactly, and every method is tied to a form the student must learn to recognize. Topics: Separable Equations, First-Order Linear Equations, Exact Equations, Solutions by Substitution, Choosing a First-Order Method.

First-Order Models

Here the solving techniques meet real situations, and each family of word problems has its own schema. Topics: Building First-Order Models, Growth, Decay, and Continuous Interest, Newton's Law of Cooling, Mixing and Compartment Models, Nonlinear Population and Reaction Models, Motion Against Resistance, Series Circuits, Families of Curves and Orthogonal Trajectories.

Numerical Methods

Most differential equations that come out of real problems have no formula solution, so this unit builds the habit of producing numbers instead. Topics: Euler's Method, Improved Euler, Runge-Kutta, and Multistep Methods, Error, Step Size, and Stability, Numerical Solutions of Systems.

Linear Second-Order Equations

Second-order linear equations are the workhorse of the subject, and this unit builds both the theory that says what a solution set looks like and the techniques that produce one. Topics: Structure of Linear Second-Order Equations, Constant-Coefficient Homogeneous Equations, Reduction of Order, Cauchy-Euler Equations.

Nonhomogeneous Linear Equations

When a linear equation has a forcing term on the right side, its general solution is the complementary function plus any one particular solution, and this unit is about producing that particular solution. Topics: Undetermined Coefficients, The Annihilator Approach, Variation of Parameters, Green's Functions.

Oscillations and Circuits

The linear second-order machinery put to work on the two systems it was built for: a mass hanging on a spring and a series circuit containing a resistor, an inductor, and a capacitor. Topics: Free Undamped Motion, Free Damped Motion, Forced Motion: Transient and Steady State, Resonance and Frequency Response, Series Circuits and the Mechanical Analogy, Other Small-Oscillation Models.

The Laplace Transform

A method that converts an initial-value problem into an algebra problem. Topics: The Transform and Its Existence, Inverse Transforms, the s-Shift, and Poles, Solving Initial-Value Problems by Transform, Step Functions and Piecewise Forcing, Impulse Forcing, Further Operational Properties, Convolution, Transfer Functions, and Impulse Response, Systems by the Laplace Transform.

Linear Systems

Several unknown functions of the same variable, coupled together and written as one matrix equation X' = AX + F. Topics: Systems in Matrix Form, Solving Systems by Elimination, Distinct Real Eigenvalues, Complex Eigenvalues, Repeated Eigenvalues, Phase Portraits and Stability, Nonhomogeneous Systems, Fundamental Matrices and the Matrix Exponential, Systems from Applications.

Nonlinear Systems and Stability

Almost no nonlinear system of differential equations can be solved with a formula, so this unit answers a different question: what do the solutions do? The plane is organized around the critical points, the places where both rates are zero and the system sits still. Topics: Critical Points and Exact Trajectories, Linearization and Almost-Linear Stability, Interacting Populations, The Nonlinear Pendulum and Conservative Systems, Periodic Solutions and Long-Term Behavior.

Series Solutions

When the coefficients of a linear second-order equation are not constants, the characteristic-equation machinery fails and there is usually no elementary formula for the solution. Topics: Power Series Tools, Series Solutions at an Ordinary Point, Regular Singular Points and Frobenius Series, Bessel and Legendre Equations.

Fourier Series and Boundary Value Problems

Everything so far has fixed the arbitrary constants with conditions at a single point. Topics: Eigenvalue Problems, Boundary-Value Models and Numerical Solution, Fourier Series, The Heat Equation, The Wave Equation, Laplace's Equation, Sturm-Liouville Problems and Eigenfunction Expansions.