A Treatise on Ordinary and Partial Differential Equations

Abstract

WE have read Prof. Woolsey Johnson's work with some interest: his style is clear, and the worked-out examples well adapted to elucidate the points the writer wishes to bring out. He appears to recognize Boole, but, so far as the text is concerned, does not acknowledge the existence of Mr. Forsyth's fine work. We do not say that he was under any obligation to do so, but nowadays we are so accustomed to see a list of authors upon whom a writer has drawn that we missed it here. “An amount of space somewhat greater than usual has been devoted to the geometrical illustrations which arise when the variables are regarded as the rectangular co-ordinates of a point. This has been done in the belief that the conceptions peculiar to the subject are more readily grasped when embodied in their geometric representations. In this connection the subject of singular solutions of ordinary differential equations, and the conception of the characteristic in partial differential equations may be particularly mentioned.” This is certainly the most prominent feature of the early chapters, and it is, to our mind, clearly put before the student. Reference is duly made to Prof. Cayley's work in the Messenger of Mathematics (vol. ii.), which initiated the present mode of treatment of the subject, but not to Dr. Glaisher's “Illustrative Examples” (vol. xii.), nor to Prof. M. J. M. Hill's paper (London Math. Soc. Proc., voL. xix.), in which the theorems stated by Prof. Cayley are proved. This paper, though read before the Society, June 14, 1888, may not have reached the author before his work was in the printer's hands: we do not say that a perusal of it would have called for any further notice than a reference. Symbolic methods come in for their due meed of recognition and employment. The author satisfies himself with referring the student to the table of contents for the topics included and the order pursued in treating them. The work consists of twelve chapters divided up into twenty-four sections: i. (i) discusses the nature and meaning of a differential equation between two variables; ii. (2, 3, 4,) equations of the first order and degree; iii. equations of the first order, but not of the first degree, (5) singular solutions (discriminant, cusp-, tac-, and node-loci), (6) Clairaut's equation, (7) geometrical applications, orthogonal trajectories; iv. (8) equations of the second order; v. (9, 10) linear equations with constant coefficients, in (10) symbolic methods are employed; vi. (11-13) linear equations with variable coefficients; vii. (14, 15) solutions in series; viii. (16) the hypergeometric series; ix. (17) special forms of differential equations, as Riccati's equation (due reference is made to Dr. Glaisher's classical paper in the Phil. Trans. for 1881.), Bessel's equation, and Legendre's equation (reference is made to text-books and memoirs); x. (18-20) equations involving more than two variables; xi. (21, 22) partial differential equations of the first order; xii. (23, 24) partial differential equations of higher order. Examples for practice are added at the end of each section. Though Prof. Johnson cannot lay claim to have made here any additions to our knowledge of the subject, he has produced an excellent introductory hand-book for students, and this, we expect, was the object he proposed to himself in its compilation. We have omitted to state that all use of the complex variable is eschewed.