Solve y−2px+ayp²=0

The equation y − 2px + ayp² = 0 (p = dy/dx) is solvable for x.

Solve for x

2px = y + ayp² → x = y(1 + ap²)/(2p) = y(1/(2p) + ap/2)

Differentiate w.r.t. y

dx/dy = 1/p, so

1/p = (1 + ap²)/(2p) + y(−1/(2p²) + a/2) dp/dy

1/p − (1 + ap²)/(2p) = (1 − ap²)/(2p), and −1/(2p²) + a/2 = (ap² − 1)/(2p²). So

(1 − ap²)/(2p) = y (ap² − 1)/(2p²) · dp/dy

(1 − ap²)[1/(2p) + y/(2p²) · dp/dy] = 0

General solution

From the second factor: 1/p + (y/p²) dp/dy = 0 → dp/p = −dy/y, so py = c, p = c/y.

Substitute in the given equation:

y − 2(c/y)x + ay(c²/y²) = 0 → y² − 2cx + ac² = 0

y² = 2cx − ac²

Singular solution

The first factor gives ap² = 1, p = ±1/√a. Then the equation reduces to 2y = 2px, y = px = ±x/√a, so

y² = x²/a

This is also the envelope of the family: ∂/∂c gives 2x − 2ac = 0, c = x/a, and y² = 2x²/a − x²/a = x²/a.