Theory of Differential Equations Volume 5; (vol. I) Exact equations and Pfaff's problem. 1890

9781236217585: Theory of Differential Equations Volume 5; (vol. I) Exact equations and Pfaff's problem. 1890

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. 1906 Excerpt: ...a new variable z' by the relation z'= z--px, and assigning other associated variables in the form y' = y, i' = q, »'=-p, p'=a. The quantity p'P' + q'Q' which occurs in the modified system is obtained from w(X + pZ) + qQ, by making the above substitutions; this does not vanish in general, and so the process can be applied to the modified equation: the integral of the original equation can be deduced as before. Note 1. Both in Cauchy's method and in Darboux's modification, an integral has been obtained which has been declared a general integral. Its expression certainly contains an arbitrary function in the least restricted case; but this property is not, of itself, sufficient to secure the character in the customary form. A general integral is given by the elimination of a between the equations gx, y, z, a,/(a) = 0, that is, one of the equations is the derivative of the other with regard to the parameter which is to be eliminated. Of course, when the elimination can be actually achieved, this relation between the two equations disappears; usually, however, the expression of the general integral.must be left in this form. In particular instances, the result can be verified by bringing the last two equations into an equivalent form which exhibits the relation characteristic of the general integral. Note 2. The general integral that has thus been obtained is the integral specified in Cauchy's existence-theorem. Taking the simpler form, we have an integral such that y = yo and z = z0 = f (y0), when x = x0, where f can be any function subject to the conditions involved in the existence-theorem for ordinary differential equations: in other words, the integral is such that z acquires a value j (y), when x=x0. The conditions have relation to the regularity of t...

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