Book: A Treatise on Differential Equations

Scope and Aim
George Boole's 1859 A Treatise on Differential Equations offers a systematic, algebraic exposition of methods for solving both ordinary and partial differential equations. Emphasis falls on linear equations and the treatment of variable coefficients, but substantial attention is given to nonlinear cases, singular solutions, and the reduction of problems to quadratures. The book aims to unify techniques under a clear symbolic framework that highlights general principles as well as practical procedures.
The treatise addresses equations of first and higher order, simultaneous systems, and the foundations of integration methods that recur in applied mathematics. Examples drawn from geometry and physical problems illustrate the theoretical developments, while auxiliary discussions clarify the conditions under which particular solution methods succeed.

Core Theory
A central theme is the structure and solvability of linear differential equations. Boole develops the theory of linear operators and relations among solutions, making explicit the role of homogeneous equations, superposition, and the determination of complementary functions. He treats equations with constant and variable coefficients, showing how transformations and reductions can simplify higher-order problems to lower-order forms.
Singular integrals and singular solutions receive careful treatment. Boole examines envelopes of general integrals and conditions that give rise to singular solutions, reconciling algebraic and geometric viewpoints. The consideration of existence and uniqueness is practical rather than axiomatic, aimed at guiding computation and interpretation of results.

Methods and Techniques
Techniques are presented with algebraic clarity: integrating factors, reduction of order, series expansions, and methods of undetermined coefficients are developed and illustrated. Boole demonstrates how to convert differential relations into integrable forms and how substitutions tailored to the equation's structure produce reductions to quadratures. Operational manipulation of differentials and systematic use of symbolic notation make these procedures repeatable across families of equations.
Series methods for solutions near ordinary and singular points are used to tackle variable-coefficient problems, with attention to convergence and the behavior of term-by-term integration. The book balances algorithmic recipes for obtaining explicit integrals with broader insights about when such recipes will succeed.

Partial Differential Equations
First-order partial differential equations are treated by classical methods for obtaining complete integrals, including the method of characteristics and parametric integration. Boole explores the reduction of certain classes of partial equations to total differential forms and the construction of integrals by means of appropriate substitutions and auxiliary conditions.
Higher-order and linear partial equations are discussed in the context of superposition and separation of variables where applicable, and the text connects techniques for ordinary differential equations to analogous procedures for partial problems, emphasizing the transfer of ideas between the two realms.

Style and Legacy
Boole writes in a compact, rigorous style that blends algebraic manipulation with geometric intuition. The exposition favors general principles and symbolic economy, which makes the treatise especially useful for readers seeking a coherent framework rather than a catalog of isolated tricks. Worked examples and explanatory remarks make the methods accessible to students with a solid grounding in calculus and algebra.
The treatise influenced the 19th-century development of differential equation theory by stressing systematic methods and operator thinking that later matured into more formal operator calculus and functional analysis. Its combination of technique, theory, and illustrative application ensured that the book remained a significant reference for mathematicians and practitioners working with differential equations.