Charpit's Method Calculator

Solve non-linear first-order PDEs f(x, y, z, p, q) = 0 by Charpit's method: the auxiliary equations, and the complete integral for the four standard forms, checked by substituting back.

Equation in x, y, z, p, q	p^2 + q^2 = 1

Equation in x, y, z, p, q

p = ∂z/∂x, q = ∂z/∂y. Examples: pq = 1, z = px + qy + pq, p(1 + q) = qz, p^2 - x = q^2 + y.

Complete integral

z = a·x + y·√(−(a^2 − 1)) + c

This is standard form I: f(p, q) = 0. The complete integral is z = a·x + y·√(−(a^2 − 1)) + c, with arbitrary constants a and c. Taking the other square root gives a second family.

Exact form. Not independently checked.

Method: Charpit's method (standard forms)
Worked out with: AllCalcHub expression engine
Check: Every step that can be checked was checked

Form: Standard form I: f(p, q) = 0
Charpit's equations: dp/(0) = dq/(0) = dz/(−(2p^2 + 2q^2)) = dx/(−2p) = dy/(−2q)

About the Charpit's Method Calculator

Charpit's method looks for a second relation between p and q from the auxiliary equations dp/(F_x + pF_z) = dq/(F_y + qF_z) = dz/(−pF_p − qF_q) = dx/(−F_p) = dy/(−F_q), then integrates dz = p dx + q dy.

The standard forms make that relation obvious: f(p, q) = 0 (try z = ax + by + c); Clairaut's z = px + qy + g(p, q) (z = ax + by + g(a, b)); f(z, p, q) = 0 (put u = x + ay); and f₁(x, p) = f₂(y, q) (set both equal to a).

The formula

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

Worked example

p² + q² = 1

z = ax + by + c with a² + b² = 1
z = ax + √(1 − a²) y + c

Things to keep in mind

The complete integral is found for the four standard forms (with p or q at most quadratic); other equations get Charpit's auxiliary equations to work by hand.

Questions people ask

What is a complete integral?

A solution with as many arbitrary constants as independent variables (two here). Others, like the singular solution, come from it.

Why only one square root?

Quadratic equations give two families; the other uses the opposite sign of the root.