An elementary treatise on differential equations and their applications . Fig. 12. Zy = 2px-2±-. 3? Ex. (ii). Proceeding as in the last chapter, we get px2 - 2p2 = (2X3 - ipx) dpdx .(A) xi-2p=0 or p = 2x-~- dx dp— =2—,x p * We say in general, because it is conceivable that in some special example anode- or cusp-locus may coincide with an envelope or with a curve of the family. 70 DIFFERENTIAL EQUATIONS log x = 2 log p - log c, whence 3«/ = 2cV-2c, (3«/ + 2c)2 = 4cx3, a family of semi-cubical parabolas with their cusps on the axis of y. The c-discriminant is (3y -x3)2 = 9y2,
An elementary treatise on differential equations and their applications . Fig. 12. Zy = 2px-2±-. 3? Ex. (ii). Proceeding as in the last chapter, we get px2 - 2p2 = (2X3 - ipx) dpdx .(A) xi-2p=0 or p = 2x-~- dx dp— =2—,x p * We say in general, because it is conceivable that in some special example anode- or cusp-locus may coincide with an envelope or with a curve of the family. 70 DIFFERENTIAL EQUATIONS log x = 2 log p - log c, whence 3«/ = 2cV-2c, (3«/ + 2c)2 = 4cx3, a family of semi-cubical parabolas with their cusps on the axis of y. The c-discriminant is (3y -x3)2 = 9y2, aj3(6!/-x3)=0. The cusp-locus appears cubed, and the other factor represents theenvelope. It is easily verified that 6y = x3 is a solution of the differentialequation, while x=0 (giving p — cc ) is not. If we take the first alternative of the equations (a), x* ?2?=0, we get by substitution for p in the differential equation Zy. * the envelope. This illustrates another method of finding singula* Fig. 13. « Examples for solution. Find the complete primitives and singular solutions (if any) of thefollowing differential equations. Trace the graphs for Examples 1-4:(1) ip2-2x = 0. (2) 4^2(z-2)=l. (3) xp2-2yp + ix=0. (4) p2 + y2-l=0. (5) p2 + 2xp-y = 0. (6) xp2-2yp + l=0. (7) ixp2 + iyp-l=0. SINGULAR SOLUTIONS 7i 59. The p-discriminant. We shall now consider how to obtainthe singular solutions of a differential equation directly from theequation itself, without having to find the complete primitive. Consider the equation x2p2 - yp +1 =0. If we give x and y any definite numerical values, we get a quad-ratic for p. For example, if x = ^/2, «/ = 3, 2y2-3^ + l=0, V**\ or !• Thus there are two curves of the family satisfying this equationthrough every point. These two curves will have the same tangentat all points where the equation has equal roots in p, wherethe discriminant y2 - ix2 = 0. Similar conclusions hold for the quadratic Lp2+Mp+N =0,w
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