An elementary treatise on differential equations and their applications . 64. Summary of results. The ^-discriminant therefore may beexpected to contain (i) the envelope,(ii) the tac-locus squared,(iii) the cusp-locus,and the c-discriminant to contain (i) the envelope,(ii) the node-locus squared,(iii) the cusp-locus cubed. SINGULAR. SOLUTIONS 75 Of these only the envelope is a solution of the differentialequation. 65. Examples. Ex.(i). j)»(2-3y)» = 4(l-ff). Writing this in the form d£_ 2-3yd!r±2s(l-y)we easily find the complete primitive in the form (xrc)* = y*(L-y).The c-discriminant. and ^-d
An elementary treatise on differential equations and their applications . 64. Summary of results. The ^-discriminant therefore may beexpected to contain (i) the envelope,(ii) the tac-locus squared,(iii) the cusp-locus,and the c-discriminant to contain (i) the envelope,(ii) the node-locus squared,(iii) the cusp-locus cubed. SINGULAR. SOLUTIONS 75 Of these only the envelope is a solution of the differentialequation. 65. Examples. Ex.(i). j)»(2-3y)» = 4(l-ff). Writing this in the form d£_ 2-3yd!r±2s(l-y)we easily find the complete primitive in the form (xrc)* = y*(L-y).The c-discriminant. and ^-discriminant are respectively y*(l-y)=0 and (2-3y)*(l-y)= - y=0, which occurs in both to the first degree, gives an envelope ;y=0, which occurs squared in the c-discriminant and not at all inthe p-discriniinant, gives a node-locus ; 2 -3y=0, which occurs squaredin the ^-discriminant and not at all in the c-discriminant, gives atac-locus. It is easily verified that of these three loci only the equation of theenvelope satisfies the differential equation. Ehuelope. FlO. 20. Ex. (ii). Consider the family of circles z2 + j/2 + 2cz+2c2-l= eliminating c (by the methods of Chap. I.), we obtain the differ-ential equation 2y2p2 + Ixyp + x2 + y2 -1 = 0. 76 DIFFERENTIAL EQUATIONS The c- and ^-discriminants are respectively x2-2(x2 + y2-l)=0 and x2y2-2y2(x2 + y2-l)=0, x2 + 2y2-2=0 and y2(x* + 2y2-2)=0. x2 + 2y2-2=0 gives an envelope as it occurs to the first degree inboth discriminants, while y = 0 gives a tac-locus, as it occurs squaredin the y-discriminant and not at all in the c-discriminant.
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