An elementary treatise on differential equations and their applications . 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-d
An elementary treatise on differential equations and their applications . 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 FIQ. 21. Examples for solution. In the following examples find the complete primitive if the differ-ential equation is given or the differential equation if the completeprimitive is given. Eind the singular solutions (if any). Trace thegraphs. (1) ±x(x-l){x-2)p2-(3x2-6x + 2)2=*0. (2) ixp2-(3x-l)2=0. (3) yp2-2xp + y = 0. (4) 3xp2-6yp + x + 2y=0. (5) p2 + 2px*-ix2y=0. (6) p3-4:xyp + 8y2=0. (7) x2 + y2-2cx + c2coa2a = 0. (8) c2 + 2cy-x2 + l =0. (9) c2 + (x + y)c + l-xy=0. (10) x2 + y2 + 2cxy + c2-l =0. 66. Clairauts Form.* We commenced this chapter by con-sidering the equation a y=px+-. * Alexis Claude Clairaut, of Paris (1713-1765), although best known in con-nection with differential equations, wrote chiefly on astronomy. SINGULAR SOLUTIONS 77 This is a particular case of Clairauts Form y=px+f(p) (1) To solve, differentiate with respect to y. p=P + {x+f(p)}-£\ therefore dp =0, p- dx~ *~^ (2) or 0=x+/(p) (3) Using (1) and (2) we get the complete primitive, the family ofstraight lines,
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