An elementary treatise on differential equations and their applications . To sum up, we may expect the c-discriminant to contain:(i) the envelope,(ii) the node-locus squared,(iii) the cusp-locus cubed. SINGULAR SOLUTIONS 69 The envelope is a singular solution, but the node- and cusp-loci are not (in general *) solutions at all. 58. The following examples will illustrate the preceding results : Ex. (i). y^p\ The complete primitive is easily found to be iy = (x-c)2, c2 - 2cx + x2 - 4«/ = 0. As this is a quadratic in c, we can write down the discriminant atonce as (2a;)2 = 4(a;2-4?/), «/
An elementary treatise on differential equations and their applications . To sum up, we may expect the c-discriminant to contain:(i) the envelope,(ii) the node-locus squared,(iii) the cusp-locus cubed. SINGULAR SOLUTIONS 69 The envelope is a singular solution, but the node- and cusp-loci are not (in general *) solutions at all. 58. The following examples will illustrate the preceding results : Ex. (i). y^p\ The complete primitive is easily found to be iy = (x-c)2, c2 - 2cx + x2 - 4«/ = 0. As this is a quadratic in c, we can write down the discriminant atonce as (2a;)2 = 4(a;2-4?/), «/ = 0, representing the envelope of the family of equal parabolasgiven by the complete primitive, and occurring to the first degree only,as an envelope 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* solutions.
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