Lagrange's method for a PDE

For Pp + Qq = R the auxiliary equations are dx/P = dy/Q = dz/R. Here P = y², Q = −xy, R = x(z − 2y).

dx/y² = dy/(−xy) = dz/[x(z − 2y)]

First integral. Take the first two ratios: dx/y² = dy/(−xy), so −x dx = y dy.

x dx + y dy = 0 → x² + y² = a

Second integral. Take the last two ratios (cancel x from the third): dy/(−y) = dz/(z − 2y).

dz/dy = −(z − 2y)/y      →     dz/dy + z/y = 2
Multiply by y:           y dz/dy + z = 2y   →   d(yz)/dy = 2y
Integrate:               yz = y² + b        →   yz − y² = b

General solution

φ( x² + y², yz − y² ) = 0

where φ is an arbitrary function. Equivalently, yz − y² = f(x² + y²).