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By V. I. Smirnov and A. J. Lohwater (Auth.)

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Let the function y and a certain number of consecutive derivatives of y:y',y", . . , y**""1*, be excluded from the equation, which has t h e form: &(xyyW,y(k+1\.. ,yW) = 0. ,4n-k>) thus lowering the order = O. ,Cn-k)f which we discussed in [15]. 2. e. ,v(n))==o, we take y as independent variable and introduce the new function V =

The equation of the envelope of family (IS) can be obtained by eliminating G from the two equations: V(x,y,C) = 0; » < y ° > =0. (82) As we move along the envelope, we touch different curves of family (78), each curve being defined by its value of constant C; this makes it clear why the equation of the envelope was sought in the form (78), with 0, however, taken as variable. We now turn to the singular solution of a differential equation. e. the coordinates >(x, y) and slope y' of the tangent for any given curve of family (78) satisfy equation (83).

On substituting we get: a; J (u'-fw 2 ) = (1 — xu)2, which gives us the linear equation for u: u' + — X u - \ 2= 0 . X Integration of this latter gives: u = ar 2 (Cx + x)=Cl x~2 + ar 1 . On substituting for u in the expression for y, we get: y = e-Clx-i+ logx + C or y = CtxeCLx-if c where we have written <72 = e and replaced (—C7X) by Cv 18. Systems of ordinary differential equations. A system oi n first order equations with n unknown functions has the form, on solving with respect to the derivatives: Ax = /i(*.

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