Solve p³−4xyp+8y²=0

p³ − 4xyp + 8y² = 0, with p = dy/dx. It is solvable for x.

Solve for x

4xyp = p³ + 8y² → x = p²/(4y) + 2y/p

Differentiate w.r.t. y

dx/dy = 1/p:

1/p = p/(2y) · dp/dy − p²/(4y²) + 2/p − (2y/p²) dp/dy

Collect the terms:

0 = (dp/dy)[p/(2y) − 2y/p²] − p²/(4y²) + 1/p

p/(2y) − 2y/p² = (p³ − 4y²)/(2yp²), and 1/p − p²/(4y²) = −(p³ − 4y²)/(4y²p). Therefore

(p³ − 4y²)[ (1/(2yp²)) dp/dy − 1/(4y²p) ] = 0

General solution

The second factor: dp/dy = p/(2y), so dp/p = dy/(2y), giving p² = cy.

Substitute p³ = p · cy into the given equation:

p(cy − 4xy) + 8y² = 0 → p(c − 4x) = −8y → p = 8y/(4x − c)

Put this in p² = cy: 64y²/(4x − c)² = cy, so

64y = c(4x − c)²

(Check: p = c(4x − c)/8 and p² = c²(4x − c)²/64 = cy ✓.)

Singular solution

The first factor p³ = 4y² with the equation gives 12y² = 4xyp, so p = 3y/x. Then 27y³/x³ = 4y², so

y = 4x³/27

It also satisfies the differential equation, and it is the envelope of the family 64y = c(4x − c)².