Notes on Diffy Qs: Differential Equations for Engineers

Citation

Lebl, J. (2026). Notes on Diffy Qs: Differential Equations for Engineers (Edition 6, version 6.11). Jiří Lebl. Free official edition: https://www.jirka.org/diffyqs/

Chapter Summary

Introduction

The opening chapter has an instructor-facing section on the book's origins (Math 286 at the University of Illinois), chapter dependencies and sample course plans, and then introduces differential equations for students: what a solution is, how modelling moves from a real problem to an equation and back, four "fundamental equations" whose solutions are worth memorising (exponential growth and decay and the two basic oscillation equations), and the classification of equations as ordinary or partial, linear or nonlinear, and by order.

Chapter 1: First-order equations

This chapter covers the standard first-order toolkit: integrating to find solutions, slope fields together with Picard's theorem on existence and uniqueness, separable equations (including implicit solutions), linear equations solved with an integrating factor, and substitutions such as Bernoulli and homogeneous equations. It then turns to qualitative analysis of autonomous equations (equilibria, stability and the logistic equation), Euler's method for numerical approximation, exact equations with integrating factors, and a short optional section on first-order linear PDEs. Each section opens with a note on lecture time and whether it can be skipped.

Chapter 2: Higher-order linear ODEs

Second-order linear equations are studied first in general (superposition, linear independence, reduction of order) and then with constant coefficients, where complex roots of the characteristic equation are handled with Euler's formula. The chapter extends this to higher-order equations, applies it to free mechanical vibrations (undamped and damped mass–spring systems, with RLC circuits as an analogue), solves nonhomogeneous equations by undetermined coefficients and variation of parameters, and ends with forced oscillations, resonance and practical resonance.

Chapter 3: Systems of ODEs

The chapter shows how higher-order equations become first-order systems, states Picard's theorem for systems, and gives a compact review of matrices, determinants and linear systems (Appendix A offers a fuller treatment). Linear systems are solved by the eigenvalue method with real, complex and repeated (defective) eigenvalues, and two-dimensional systems are visualised through vector fields and phase portraits. Further sections treat coupled mass–spring systems and normal modes, matrix exponentials and fundamental matrix solutions, and nonhomogeneous systems.

Chapter 4: Fourier series and PDEs

Starting with boundary value and eigenvalue problems (including orthogonality of eigenfunctions and the Fredholm alternative), the chapter develops trigonometric Fourier series through an inner-product analogy with eigenvector decomposition, discusses convergence and differentiation, and introduces sine and cosine series. These are applied to periodically forced oscillations and resonance and then to the classical PDEs: the heat equation by separation of variables, the one-dimensional wave equation and d'Alembert's solution, steady-state temperature and the Laplacian, and the Dirichlet problem in a disk with the Poisson kernel.

Chapter 5: More on eigenvalue problems

This chapter generalises Chapter 4: Sturm–Liouville problems, their orthogonality and eigenfunction series; higher-order eigenvalue problems illustrated by vibrating beams; and steady periodic solutions, with examples of a forced vibrating string and underground temperature oscillations.

Chapter 6: The Laplace transform

The Laplace transform is defined with its inverse and used to turn linear ODEs with initial conditions into algebraic problems. The chapter covers transforms of derivatives and integrals, the Heaviside function for switched inputs, transfer functions, periodic inputs, convolution (including a Volterra integral equation), the Dirac delta and impulse response, an example of three-point beam bending, and an optional section on solving PDEs with the transform.

Chapter 7: Power-series methods

After reviewing power series, radius of convergence and analytic functions, the chapter finds series solutions of second-order linear equations about ordinary points and then treats singular points with the method of Frobenius, ending with Bessel functions.

Chapter 8: Nonlinear systems

The final chapter studies autonomous nonlinear systems in the phase plane: critical points, linearisation, stability and classification of isolated critical points, the difficulty that linearised centres pose, and conservative equations. Applications include the pendulum and predator–prey (Lotka–Volterra) models; there are short sections on limit cycles, with the Van der Pol oscillator, and on chaos, with the Duffing equation, strange attractors, Poincaré sections and the Lorenz system.

Appendix A: Linear algebra

A self-contained linear algebra appendix covers vectors and linear maps, matrix algebra, elimination and row echelon form, rank, subspaces, basis and kernel, inner products and projections with Gram–Schmidt, and determinants. It can replace the short review in Section 3.2 for courses that combine linear algebra with ODEs.

Appendix B: Table of Laplace Transforms

A reference table of transforms used in Chapter 6, followed in the back matter by further reading and solutions to selected exercises.

Key Concepts

Introduction

• Mathematical model: translating a real-world situation into a differential equation, solving it, and interpreting the solution back in context.
• Classification: ODEs versus PDEs, linear versus nonlinear, and the order of an equation decide which methods apply.

Chapter 1: First-order equations

• Picard's theorem: if the right-hand side and its partial derivative in y are continuous near the initial point, a unique solution exists for some interval.
• Integrating factor: a multiplier that turns a first-order linear equation into an exact derivative that can be integrated.
• Autonomous equation: an equation y′ = f(y) whose equilibria and their stability can be read without solving it.

Chapter 2: Higher-order linear ODEs

• Characteristic equation: the polynomial whose roots give exponential (or, via Euler's formula, oscillating) solutions of constant-coefficient equations.
• Undetermined coefficients and variation of parameters: two methods for finding a particular solution of a nonhomogeneous equation.
• Resonance: unbounded or large-amplitude response when a forcing frequency matches (or nearly matches) a natural frequency.

Chapter 3: Systems of ODEs

• Eigenvalue method: solving x′ = Ax with solutions of the form v e^{λt} built from eigenvalues and eigenvectors of A.
• Phase portrait: a picture of trajectories of a two-dimensional system that shows qualitative behaviour.
• Matrix exponential: e^{tA}, which gives the fundamental matrix solution of a constant-coefficient system.

Chapter 4: Fourier series and PDEs

• Fourier series: representation of a periodic function as a sum of sines and cosines, with coefficients obtained from an inner product.
• Separation of variables: looking for product solutions of a PDE, which reduces it to ODE eigenvalue problems.
• D'Alembert's solution: writing solutions of the wave equation as the sum of two travelling waves.

Chapter 5: More on eigenvalue problems

• Sturm–Liouville problem: a class of boundary value problems whose eigenfunctions are orthogonal and can expand other functions.
• Steady periodic solution: the long-term periodic response of a forced system, found by eigenfunction expansions.

Chapter 6: The Laplace transform

• Laplace transform: an integral transform that turns differentiation into multiplication by s, so initial value problems become algebra.
• Convolution: the operation whose transform is the product of transforms, used to express responses to general inputs.
• Impulse response: the solution driven by a Dirac delta input, from which other responses can be built.

Chapter 7: Power-series methods

• Ordinary and singular points: points where series solutions behave regularly or require modified methods.
• Method of Frobenius: seeking solutions of the form x^r times a power series near a regular singular point, leading for instance to Bessel functions.

Chapter 8: Nonlinear systems

• Linearisation: approximating a nonlinear system near a critical point by its Jacobian matrix to classify behaviour.
• Limit cycle: an isolated closed trajectory that nearby solutions approach or leave.
• Chaos: sensitive dependence on initial conditions, illustrated with the Duffing equation and the Lorenz system.

Appendices A–B

• Kernel and rank: the solution space of Ax = 0 and the number of independent columns, linked through elimination.
• Gram–Schmidt process: turning a basis into an orthogonal one using projections.

Critical Analysis

Strengths

• Designed as a teaching tool. Every section begins with a note giving lecture time, whether it can be skipped and the matching sections in Edwards and Penney and in Boyce and DiPrima. The introduction gives a dependency chart and sample one- and two-semester plans, including a combined linear algebra and ODE course, so instructors can adapt the book easily.
• Broad coverage in one free text. Beyond standard first- and second-order methods, it covers systems with matrix exponentials, Fourier series and the three classical PDEs, Sturm–Liouville theory, Laplace transforms, series and Frobenius methods, and nonlinear dynamics through chaos—roughly the content of the two widely used commercial texts it aims to replace.
• Engineering examples throughout. Mass–spring systems, RLC circuits, coupled oscillators, beam deflection, the heat in a wire and underground temperature oscillations make the methods concrete without long modelling detours.
• Exercises and online resources. Each section has exercises; those numbered 101 and above have solutions in the back. The website provides interactive Sage demos, WeBWorK homework sets for most sections, figures and videos for some sections, and the LaTeX source under an open licence.

Limitations

• Methods first, theory light. Existence and uniqueness are stated (Picard's theorem) and used but not proved, and convergence of Fourier series is described rather than proved. Students looking for a theoretical ODE course will need a dynamical-systems or analysis-based text.
• Nonlinear dynamics is brief. Chapter 8 introduces stability, limit cycles and chaos in a few sections; Lyapunov functions and bifurcation theory are not developed.
• Numerical methods are minimal. Only Euler's method gets a section; the Runge–Kutta method appears in its exercises and is reused later for systems, but error analysis and numerical stability are not treated systematically.
• Assumes the course order. The author notes that chapters should largely be covered in order; Chapters 4–5 and parts of 6 refer back to earlier material, so readers dipping into a single later topic (for example Laplace transforms or Sturm–Liouville problems) may need to look back for notation and methods.

Applications in the Book

Applications are woven into sections rather than collected in separate chapters. They include exponential growth and Newton's law of cooling (Introduction and Chapter 1), the logistic equation, mass–spring systems and RLC circuits with free, damped and forced motion and resonance (Chapter 2), coupled mass–spring systems and normal modes (Section 3.6), periodically forced oscillations via Fourier series (Section 4.5), heat conduction, vibrating strings and steady-state temperature (Sections 4.6–4.10), vibrating beams, a forced string and underground temperature oscillations (Chapter 5), switched and impulsive inputs and three-point beam bending with the Laplace transform (Chapter 6), and the pendulum, Lotka–Volterra predator–prey, Van der Pol, Duffing and Lorenz systems (Chapter 8).

Where It Fits in the Course

Lebl's book sits alongside Boyce, DiPrima and Meade's Elementary Differential Equations and Boundary Value Problems, an undergraduate textbook for the same first course. The author states that his notes aim to be a "drop-in replacement" for Boyce and DiPrima or for Edwards and Penney, and every section notes the corresponding sections there, so the two books can be used interchangeably or side by side; Lebl is the free option, and students can cross-reference to the commercial text for extra examples.

Hirsch, Smale and Devaney's Differential Equations, Dynamical Systems, and an Introduction to Chaos is generally described as a more theoretical, linear-algebra-based treatment of dynamical systems, and is a natural next step after Lebl's Chapters 3 and 8. Arrowsmith and colleagues' Dynamical Systems and Brock and colleagues' Differential Equations, Stability, and Chaos in Dynamic Economics are graduate-level texts that suit students who want rigorous stability and chaos theory, or applications in economics, after this introductory course.

How to Use This Book

The book is written for engineering and science students taking a first course in differential equations. Single-variable calculus is essential; some multivariable calculus helps for exact equations and PDEs. Linear algebra is reviewed in Section 3.2, and Appendix A covers it fully for students without a previous course.

For a one-semester course, follow one of the plans in Section 0.1: for example the Introduction, Sections 1.1–1.7, Chapter 2, Chapter 3 and Sections 4.1–4.9, plus one of Chapters 5–8 as time allows. A course without PDEs can instead cover Chapters 1, 2, 6 and 7. Sections flagged as skippable (1.5, 1.7–1.9, 6.5) can be left for reading. Work several exercises per section and check those numbered 101 and above against the solutions at the back; use the online WeBWorK sets and Sage demos to practise and to visualise slope fields and phase portraits.

Edition and Sources

This guide is based on Edition 6 of Notes on Diffy Qs (version 6.11, dated 18 May 2026, © 2008–2026 Jiří Lebl, dual-licensed CC BY-NC-SA 4.0 and CC BY-SA 4.0), read in the free PDF from the author's official site, https://www.jirka.org/diffyqs/. It draws on the notes to instructors (organisation, dependency chart, sample courses, computer resources), the full table of contents, the introductory section and the per-section notes for every section, all subsection headings in Chapters 0–8 and Appendix A, the statement of Picard's theorem, the "four fundamental equations", and the examples and subsections on applications (Sections 2.4–2.6, 3.6, 4.5, 5.2–5.3, 6.4, 8.3–8.5).