Applied mathematicians, chemists, engineers, physicists, and others who use mathematics sometimes need to solve a differential equation. The related problem of evaluating an integral is usually simple, since tables of standard integrals are readily available, A similar tabulation of standard differential equations might be helpful. The idea is not new. As early as 1889, Professor William E. Byerly of Harvard University included "A Key to the Solution of Differential Equations" in the second edition of his text on integral calculus. Parenthetically, it is interesting to note that Peirce's "A Short Table of Integrals" first appeared in the 1881 edition of the same book. More recently, the well-known German treatise of E. Kamke (1940 and subsequent editions) has contained a collection of more than a thousand equations with their general solutions.
Unfortunately, consideration of the problem shows that it is not easy to select the standard equations. The value of an integral, in the usual case, can be presented in terms of parameters, independent of numerical values assigned to them. On the other hand, a change in sign of some term in a differential equation, the transfer of a term from numerator to denominator, or some other simple alteration in its form may convert the equation from one with a solution by elementary methods into a case with no solution in terms of known functions. In spite of these obvious difficulties, I was still of the opinion that it would be useful to have some formal scheme for solving a given equation. The result is the compromise offered in Parts I and II of this book. It is fully explained in the Introduction.