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Jiří Lebl

Notes on Diffy Qs

A first course in differential equations for science and engineering: first- and second-order ODEs, systems, Fourier series and PDEs, the Laplace transform, power-series methods, and nonlinear systems. The problems build both solution technique and a modeling instinct for when each tool applies.

Notes on Diffy Qs: Differential Equations for Engineers

247 exercises58 sections

Status

Sections

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Introduction

9

Introduction to differential equations6

Classification of differential equations3

First order ODEs

41

Integrals as solutions7

Separable equations7

Linear equations and the integrating factor5

Autonomous equations4

Numerical methods: Euler's method4

Exact equations3

First-order linear PDEs3

Higher order linear ODEs

31

Second-order linear ODEs5

Constant-coefficient second-order linear ODEs7

Higher-order linear ODEs6

Mechanical vibrations4

Nonhomogeneous equations6

Forced oscillations and resonance3

Systems of ODEs

36

Introduction to systems of ODEs6

Matrices and linear systems4

Linear systems of ODEs4

Eigenvalue method4

Two-dimensional systems and their vector fields2

Second-order systems and applications3

Repeated eigenvalues4

Matrix exponentials5

Nonhomogeneous systems4

Fourier series and PDEs

42

Boundary value problems5

The trigonometric series4

More on the Fourier series6

Sine and cosine series5

Applications of Fourier series4

PDEs, separation of variables, and the heat equation5

One-dimensional wave equation4

D'Alembert solution of the wave equation3

Steady state temperature and the Laplacian2

Dirichlet problem in the circle and the Poisson kernel4

Eigenvalue problems

6

Sturm--Liouville problems2

Higher-order eigenvalue problems2

Steady periodic solutions2

The Laplace transform

22

The Laplace transform4

Transforms of derivatives and ODEs4

Dirac delta and impulse response5

Solving PDEs with the Laplace transform5

Power series methods

13

Series solutions of linear second-order ODEs3

Singular points and the method of Frobenius4

Nonlinear systems

15

Linearization, critical points, and equilibria3

Stability and classification of isolated critical points4

Applications of nonlinear systems3

Linear algebra

32

Vectors, mappings, and matrices5

Matrix algebra5

Subspaces, dimension, and the kernel3

Inner product and projections7

Adapted from Notes on Diffy Qs © 2008–2026 Jiří Lebl. Licensed under CC BY-SA 4.0. Source: Notes on Diffy Qs: Differential Equations for Engineers. Imported at revision 658bcae.