Solve (x−a)p²+(x−y)p−y=0

(x − a)p² + (x − y)p − y = 0, with p = dy/dx.

Reduce to Clairaut's form

Put u = x + y, so du/dx = 1 + p, p = u' − 1. Substituting:

(x − a)(u' − 1)² + (2x − u)(u' − 1) − (u − x) = 0

Expanding and simplifying gives

(x − a)u'² + (2a − u)u' − a = 0

Divide by u' and rearrange:

u = (x − a)u' + 2a − a/u'

Let X = x − a and w = u − 2a = x + y − 2a (so dw/dX = u' = P). Then

w = XP − a/P, with P = dw/dX

This is Clairaut's equation.

General solution

Replace P by c: w = cX − a/c, that is

x + y − 2a = c(x − a) − a/c

(Equivalently y = (c − 1)x − a(c − 1)²/c, a family of straight lines. Check: with p = m, the original equation holds for y = mx − am²/(m + 1).)

Singular solution

Differentiate w = cX − a/c with respect to c: 0 = X + a/c², so c² = −a/X. Then a/c = −cX, w = 2cX and w² = 4c²X² = −4aX:

(x + y − 2a)² = −4a(x − a)

This parabola is the envelope of the family.