📄 Executive Summary
Differential Equations: An Interactive Introduction serves as an introductory college textbook designed for undergraduate students in mathematics, engineering, and the physical sciences. The text assumes a standard foundation in differential and integral calculus, along with familiarity with basic linear algebra concepts such as vectors and matrices. The book concentrates on ordinary differential equations, systematically developing theoretical principles, analytical solution techniques, numerical approximations, and qualitative geometric analyses, while deliberately leaving partial differential equations outside its scope.
The curriculum progresses logically from foundational first-order equations to higher-order equations and multidimensional linear systems. Readers begin by examining first-order differential equations and initial value problems, studying analytical methods such as separation of variables, integrating factors, substitutions for homogeneous and Bernoulli equations, and exact differentials, accompanied by existence and uniqueness theorems. The text then transitions to numerical techniques—including Euler, improved Euler, and Runge-Kutta methods—paired with error analysis and spreadsheet implementations. Subsequent chapters cover second-order and higher-order linear differential equations, investigating fundamental solution sets, the Wronskian, Abel's formula, characteristic equations, undetermined coefficients, reduction of order, and variation of parameters.
To handle broader classes of differential equations, the book introduces power series methods and integral transforms. It presents series expansions around ordinary points and the Frobenius method for equations with regular singular points. A thorough treatment of Laplace transforms follows, demonstrating how to solve initial value problems subject to discontinuous inputs, unit step functions, convolution integrals, and impulsive forces. The theoretical framework culminates in first-order linear systems of differential equations, where matrix algebra, eigenvalues, eigenvectors, and phase plane geometry are employed to classify trajectories and analyze stability.
Throughout the text, mathematical theory is connected to physical models and computational practice. Readers analyze practical applications ranging from population dynamics, Newton's law of cooling, and mixing tanks to damped and forced harmonic oscillators, RLC electrical circuits, planetary orbits under central forces, and pendulum experiments. Computational exercises incorporating spreadsheets and the open-source Sage computer algebra system provide direct experience with modern software. After studying this textbook, readers will be able to formulate mathematical models from physical situations, select appropriate analytical, series, or transform methods, carry out numerical simulations, and interpret the long-term qualitative behavior of dynamic systems.
