# Ordinary Differential Equations

1*1. Definitions.-The term oeguatio differentialis or differential equation was first used by Leibniz in 1676 to denote a relationship between the differentials dx and dy of two variables x and y* Such a relationship, in general, explicitly involves the variables x and y together with other symbols a, b, c, . . . which represent constants.

This restricted use of the term was soon abandoned ; differential equations are now understood to include any algebraical or transcendental equalities which involve either differentials or differential coefficients. It is to be understood, however, that the differential equation is not an identity.f

Differential equations are classified, in the first place, according to the number of variables which they involve. An ordinary differential equation expresses a relation between an independent variable, a dependent variable and one or more differential coefficients of the dependent with respect to the independent variable. A partial differential equati

**Table of Contents**

1 introductor* 3; ii Elementary Methods of Integration 16; III, The Existence and Nature of Solutions ok Ordinary Differential; Equations 02; IV Continuous Transformation-Groups 93; V The General Theory of Linear Differential Equations 114; VI Linear Equations with Constant Coefficients 133; /1; VII The Solution of Lin bar Differential Equations in an Infinite; Form 158; VIT! The Solution of Linear Differential Equations by Definite; integrals 186; IX The Algebraic Theory of Linear Differential yístems 204; X The Sturmian Theory and its Later Developments 223; XI Further Developments in the Theory of Boundary Problems 254; PAKT II; DIFFERENTIAL EQUATIONS IN THE COMPLEX DOMAIN; XII Existence Theorems in the Complex Domain 281; XIII, Equations of the First Order but not of the First Degree 304; XIV Non-Linear Equations of Higher Order 317; XV Linear Equations in the Complex Domain 356; XVI The So

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