Solve y = yp²+2px

The equation y = yp² + 2px, with p = dy/dx, is solvable for x, so differentiate with respect to y (treating x as a function of y).

Solve for x

2px = y − yp² → x = y(1 − p²)/(2p) = y(1/(2p) − p/2)

Differentiate w.r.t. y

dx/dy = 1/p, so

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

1/p − (1 − p²)/(2p) = (1 + p²)/(2p), so

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

Cancel (1 + p²)/(2p) (it is non-zero for real p):

1 = −(y/p) dp/dy → dy/y + dp/p = 0

Integrating: ln y + ln p = ln c, so yp = c, p = c/y.

Eliminate p

Substitute p = c/y in the given equation:

y = y(c²/y²) + 2(c/y)x → y² = c² + 2cx

General solution: y² = 2cx + c² (c an arbitrary constant).

Check: differentiating, 2yy' = 2c, so p = c/y; then yp² + 2px = c²/y + 2cx/y = (c² + 2cx)/y = y ✓.