A treatise on differential equations

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1872 Excerpt: ...primitive of a singular integral of a differential equation of the second order will itself be a singular solution of that equation, but a singular solution of a singular integral will in general not be a solution at all of that equation. The proof of these propositions will afford an exercise for the student. See Lagrange's Lemons sur le Calcul des Fonctions, Legem 14me of the edition of 1806, or Legon 15me of the edition of 1808. A note by Poisson on page 239 of the edition of 1808 should be consulted; it relates to the second of the above two propositions. See also Lacroix, Traits du Calcul Bifferentiel..., Tome n. pp. 382 and 390. 10. We proceed to inquire how singular integrals may be determined from the differential equation. Expressing as before the first integral involving an arbitrary constant in the form y=f(x,y,y1...ylt,c) (47), we have as the derived equation (df(x, y, y,...#„_„ c) the brackets in the second member indicating that in effecting the differentiation y, y„_2, are to be regarded as functions of x. The differential equation of the nth order is found from (48) by substituting therein, after the differentiation, for c its value in terms of x, y, yx,...y_v given by (47). The result assumes the form yn = (x,y, y,...jO (49). Hence, we have or, representing fix, y, y1...y, c), by /, dyn-t dxdcj dc' Hence, A iog%-=i (50), dx b dc K' provided that the first member be obtained from the differential equation, and the second member from one of its first integrals involving c as arbitrary constant. It is to be borne in mind that in effecting the differentiation with respect to x in the second member, we must regard y, yl,---yn-i as functions of x. Now reasoning as in Chap. vin. since a singular solution makes = 0, it makes its l...