Introduction

Solving equations has been one of the most important driving forces in the history of mathematics. Particularly well known is the problem of solving algebraic equations of order higher than four and its eventual solution by Lagrange, Ruffini, Abel and Galois. The complete answer given by the latter author provided not only the solution for the problem at hand, but also established a new field in mathematics, the theory of groups, which in turn laid the foundation of modern algebra. A knowledgeable and detailed review of this subject and various other topics discussed below may be found in the book by Wussing [190], see also the Historical Remarks by Bourbaki [16]. Much less known is the fact that the theory of differential equations took

a similar course in the second half of the 19th century. At that time, solving ordinary differential equations (ode’s) had become one of the most important problems in applied mathematics, about 200 years after Leibniz and Newton introduced the concept of the derivative and the integral of a function. Numerous phenomena in the physical sciences were described by formulas involving differentiation and integration, and the need arose to determine the functional dependencies between the variables involved. In other words, the problem of solving a differential equation was born. In this book the phrase solution of an ode means an expression for the general solution in some function field, e.g., in terms of elementary or Liouvillian functions. The differential equation should vanish identically upon its substitution. If n is the order of the equation the general solution involving n constant parameters is searched for. It should not involve derivatives, infinite series or products. Furthermore, any numerical or graphical representation is excluded.