About This Course

This book is about ordinary differential equations: equations that tie an unknown function of one variable to its own derivatives. It is written for a reader working alone, typically a second-year student of mathematics, physics, engineering, economics, or biology, who wants three things at once: the standard solution methods, the theorems that say when those methods are legitimate, and the qualitative tools that still work when no formula exists.

You need single-variable calculus as taught in Calculus: derivatives and integrals of the elementary functions, integration by parts and partial fractions, improper integrals, and Taylor series. You need matrix algebra, determinants, and eigenvalues and eigenvectors from Linear Algebra, and arithmetic with complex numbers including Euler’s formula \(e^{i\theta}=\cos\theta+i\sin\theta\) . Chapters 3, 4, and 7 use a few facts about the convergence of series of functions from Real Analysis; each is stated with the hypotheses needed where it is used and marked once with its library identifier, and it is not proved again here. Partial derivatives appear in Chapters 2, 3, and 8 at the level of Multivariable Calculus.

The course has done its work when you can carry out each of these tasks unaided. Translate a rate law into an initial value problem with units and check a proposed solution, including the interval on which it is valid. Recognise separable, linear, exact, and Bernoulli equations and solve them. State the existence and uniqueness theorem, apply it, and exhibit what goes wrong when a hypothesis fails. Solve linear second-order equations with constant coefficients, free and forced, and explain beats, resonance, gain, and phase lag. Use the Laplace transform for discontinuous and impulsive forcing. Solve linear systems with the matrix exponential and classify planar phase portraits. Linearize a nonlinear system at an equilibrium, decide what the linearization does and does not prove, and use conserved quantities and Lyapunov functions when it proves nothing.

Three neighbouring subjects are deliberately left to other books. Equations in several independent variables belong to Partial Differential Equations. Error analysis of numerical methods belongs to Numerical Analysis; Euler’s method appears here only as a way of seeing what a direction field means. Equations driven by random noise are not treated.

Notation is the common one. The independent variable is \(t\) and is usually time; the unknown is \(y(t)\) for a single equation and a vector \(\mathbf x(t)\) for a system. Primes denote derivatives with respect to \(t\) . An interval is written \(I\) , and “on \(I\) ” always means at every point of \(I\) . Logarithms are natural and written \(\ln\) .

Each chapter ends with exercises that run from recall to proof, and the companion solutions give full arguments for some, hints for others, and bare answers for a few. Work with pencil and paper first. Several exercises ask you to find the error in a plausible argument; differential equations offer many such arguments, usually involving a lost solution, a forgotten interval, or a hypothesis that was never checked. The cumulative review at the end mixes all chapters without saying which tool each problem needs, because choosing the tool is the skill.