A Treatise on Linear Differential Equations, Vol. 1 (Classic Reprint)

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Excerpt from A Treatise on Linear Differential Equations, Vol. 1 The theory of linear differential equations may almost be said to find its origin in Fuchs's two memoirs published in 1866 and 1868 in volumes 66 and 68 of Crelle's Journal. Previous to this the only class of linear differential equations for which a general method of integration was known was the class of equations with constant coefficients, including of course Legendre's well-known equation which is immediately transformable into one with constant coefficients. After the appearance of Fuchs's second memoir many mathematicians, particularly in France and Germany, including Fuchs himself, took up the subject which, though still in its infancy, now possesses a very large literature. This literature, however, is so scattered among the different mathematical journals and publications of learned societies that it is extremely difficult for students to read up the subject properly. I have endeavored in the present treatise to give a by no means complete but, I trust, a sufficient account of the theory as it stands to-day, to meet the needs of students. Full references to original sources arc given in every case. Most of the results in the first two chapters, which deal with the general properties of linear differential equations and with equations having constant coefficients, arc of course old, but the presentation of these properties is comparatively new and is due to such mathematicians as Hermite, Jordan, Darboux, and others. All that follows these two chapters is quite new and constitutes the essential part of the modern theory of linear differential equations. The present volume deals principally with Fuchs's type of equations, i.e. equations whose integrals are all regular: a sufficient account has been given, however, of the researches of Frobenius and Thomé on equations whose integrals arc not all regular. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.