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

ISBN-10: 0080102077

ISBN-13: 9780080102078

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E. the set of two straight lines y = ± a , which we obtained pre­ viously. 2. The general solution of Clairaut's equation y = xy' + q>(y') is y = xC +

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 =

_<.. _ , + ja. x 44 [14 ORDINARY DIFFERENTIAL EQUATIONS Let s be the length of arc of the integral curve, and let a be the angle that the positive direction of the tangent forms with the positive direction of OX. We have [I, 70]: dy dx a n a ; ds cos a, dx and we obtain, on differentiating w i t h ]respect to x: 1 d22/ — dx2 cos2 a 1 da __ da? cos 2 a da ds ds dx _ 1 cos 3 a but da/ds is the curvature of the curve, as we know from [I, 71] da 1 ~ds~ = R ' da ~di~ (13) and the previous equation gives us: 1 cos 3 a R ~ &2y dx2 (14) We take R positive here, if a increases with increasing s, and negative if a decreases with increasing s.

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A Course of Higher Mathematics. Volume II by V. I. Smirnov and A. J. Lohwater (Auth.)


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