Charpit's method for a PDE

Charpit's method. For F(x, y, z, p, q) = 0 (p = ∂z/∂x, q = ∂z/∂y), find a second relation f(x, y, z, p, q) = a from the auxiliary equations

dx/F_p = dy/F_q = dz/(pF_p + qF_q) = −dp/(F_x + pF_z) = −dq/(F_y + qF_z)

then solve for p and q and integrate dz = p dx + q dy.

Given: F = 2zx − px² − 2qxy + pq = 0.

Step 1: Partial derivatives

F_p = −x² + q          F_q = −2xy + p
F_x = 2z − 2px − 2qy   F_y = −2qx      F_z = 2x

F_x + pF_z = 2z − 2px − 2qy + 2px = 2z − 2qy
F_y + qF_z = −2qx + 2qx = 0

Step 2: A simple relation

From dq/(F_y + qF_z) = dq/0, we get dq = 0, so

q = a  (constant)

Step 3: Find p

Put q = a in F = 0:

2zx − px² − 2axy + pa = 0
p (a − x²) = 2x (ay − z)
p = 2x (ay − z)/(a − x²)

Step 4: Integrate dz = p dx + q dy

dz = 2x (ay − z)/(a − x²) dx + a dy
d(z − ay) = − 2x (z − ay)/(a − x²) dx

Let u = z − ay. Then du/u = −2x dx/(a − x²) = d[ln(a − x²)], so ln u = ln(a − x²) + ln b:

z − ay = b (a − x²)

Complete integral: z = ay + b(a − x²), with arbitrary constants a and b.

Check: p = −2bx, q = a. Then 2zx − px² − 2qxy + pq = 2x(ay + ab − bx²) + 2bx³ − 2axy − 2abx = 0 ✓.