# A Treatise on Differential Equations; Supplementary Volume Volume 2

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1865 edition. Excerpt: ...then passes on to the general theory. General Theory. 2. Given an equation of the form s = f (xv xs, ... xn) av -2, ... an), the number of arbitrary constants av as, ... an involved being equal to the number of the independent variables xv xt, ... xn, we obtain by differentiation and elimination of the constants a partial differential equation of the first order. Of this the proposed equation is said to constitute a complete primitive. The form of the above process which it seems best, as throwing light upon the inverse problem of deducing the complete primitive from the partial differential equation, to employ, is the following. Let the given primitive, solved with respect to one of the arbitrary constants av be presented in the form /(-, ... xn, z, a2, ... 0 (1). Differentiating with respect to each of the independent variables we have a system of n equations of the forms These n equations enable us first to eliminate the n--1 constants a2, an, and so deduce the partial differential equation sought in the form secondly to determine the n--1 constants as functions of a;, ... xn, z, pv... pn in the forms As the system formed of these n--1 equations, together with the previous one, is merely another form of the system (2) obtained by directly differentiating the primitive, it follows that if from these equations we deduce the values of, ... pn as functions of x, ... xa, an-... an, and substitute them in the they will render that equation integrable, and its integral will be the complete primitive (1), the constant d, being regained by integration. Examining the system (3), (4) we see that the first members of all the equations which it contains are functions of asi, ... xn, z, pi, ...pn, while the second members are constants