Differential Equations: A Practical Introduction

Differential Equations: A Practical Introduction is a working tool for students in engineering, physics, chemistry, and applied mathematics who need to solve differential equations and interpret their solutions — not debate their existence.
In 134 focused pages, this book covers the full first-year ODE curriculum:

First-order equations — separable, linear, exact, Bernoulli, with integrating factors worked in SI units
Second-order linear equations — constant coefficients, undetermined coefficients, variation of parameters
Mechanical and electrical oscillations — damping regimes, resonance, beats, and the mass-spring ↔ LRC analogy in a single unified framework
Laplace transforms — as a problem-solving tool, with Heaviside steps, Dirac deltas, and the convolution theorem
Linear systems — eigenvalue method, the four canonical phase portraits, trace-determinant classification
Qualitative analysis of nonlinear systems — Lotka–Volterra, the pendulum, limit cycles, Van der Pol, the Brusselator
Numerical methods — Euler, Heun, RK4, stiffness, adaptive stepping
Appendix A — a self-contained refresher on the calculus and linear algebra the book leans on, with Picard iteration and Jordan form
Worked solutions to every odd-numbered exercise in the back

What makes this book different:

Every worked example carries physical units. A mass is in kilograms; a spring constant is in N/m; a time constant is in seconds. When you compute T(30 min) = 44 °C, you see the °C in the answer.
Every calculation ends with one sentence of physical interpretation. What does the number mean? What has the system done? You never finish a problem wondering what you just computed.
Three figures you don't get in most textbooks. A single mass-spring-dashpot schematic drawn alongside a series LRC circuit, making the unit-by-unit analogy visible. A clean four-panel phase-portrait gallery showing saddle, stable node, stable spiral, and center side by side. A stiffness comparison showing explicit Euler blow up next to backward Euler quietly working.
Exercises in four categories. Technique (practicing the method), modeling (setting up an equation from a situation), interpretation (explaining what an answer means), and graph-reading (extracting information from a figure). Odd-numbered exercises have worked solutions in the back.
Companion to Differential Geometry: A Rigorous Introduction (Aiden Sol). Use this book for fluency in ODEs; use the companion volume when you want the geometry done properly.

Who this book is for:

Undergraduate engineering and physics students in a first or second ODE course
Chemistry and biology students meeting rate equations and compartment models for the first time
Self-learners who have calculus and want to move efficiently into differential equations without drowning in ε–δ proofs
Anyone preparing for an applied-math exam who needs to consolidate technique across separable, linear, Laplace, systems, and numerical approaches in one book