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Ordinary differential equations without the theorem-proof detour.

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

About the author. Aiden Sol writes concise, practical mathematics books aimed at students who need to get things done. Other titles in the applied mathematics series include The Mathematics of Chess, The Simple Chess Opening Repertoire (for a different kind of pattern recognition), and Differential Geometry: A Rigorous Introduction.

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