Unit 6 of 6

Further Differential Equations

  • 9 lessons
  • Parts 85 to 102

This unit continues the differential equations that close the calculus course. It solves exact first-order equations, brings in complex numbers and complex exponentials, and then treats second-order linear equations, homogeneous and nonhomogeneous.

Oscillations, damping and resonance show these equations at work. Series solutions and Laplace transforms handle equations that resist the basic methods, and the unit ends with linear systems solved by eigenvalues and nonlinear systems studied in the phase plane.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Exact Equations and Complex Numbers

  1. Exact Differential Equations

    How to tell when M dx + N dy = 0 is the differential of a function F, how to build that potential, how to repair a near miss with an integrating factor, and why the domain matters.

    Theory part 85Practice part 86

  2. Complex Numbers and Complex Exponentials

    How complex numbers are added, multiplied and divided, how polar form turns multiplication into rotation, why Euler's formula holds, and how complex exponentials produce real oscillating solutions.

    Theory part 87Practice part 88

Second-Order Linear Equations

  1. Second-Order Linear Equations

    Why a second-order linear homogeneous equation has a two-parameter family of solutions, how the Wronskian tests independence, and how the characteristic roots give every solution when the coefficients are constant.

    Theory part 89Practice part 90

  2. Nonhomogeneous Linear Equations

    How every solution of a forced linear equation splits into a homogeneous part and one particular solution, and how to find that particular solution by trial or by variation of parameters.

    Theory part 91Practice part 92

  3. Oscillations, Damping, and Resonance

    How Newton's law turns a spring, a damper and an applied force into one second-order equation, what its free, damped and forced solutions look like, and exactly when resonance occurs.

    Theory part 93Practice part 94

Series and Laplace Methods

  1. Series Solutions of Differential Equations

    How to solve linear equations with variable coefficients by power series, how far those series are guaranteed to converge, and how the Frobenius method handles singular points such as those in the Bessel equation.

    Theory part 95Practice part 96

  2. Laplace Transforms and Initial-Value Problems

    How the Laplace transform turns a linear initial-value problem into algebra, for which s it converges, and how steps, impulses and convolution handle switched and sudden inputs.

    Theory part 97Practice part 98

Systems of Equations

  1. Linear Systems and Eigenvalues

    How to write coupled linear equations as one matrix equation, solve it with eigenvalues, eigenvectors and the matrix exponential, and read off whether solutions decay, grow or circle.

    Theory part 99Practice part 100

  2. Nonlinear Systems and Phase-Plane Models

    How to find the equilibria of a nonlinear system, classify them with the Jacobian when they are hyperbolic, and use conserved quantities or radial equations when the linear picture says nothing.

    Theory part 101Practice part 102