Lagrange Linear PDE Calculator
Solve Lagrange's linear PDE Pp + Qq = R from its auxiliary equations dx/P = dy/Q = dz/R, giving the general solution φ(u, v) = 0, with u and v checked by substituting into the PDE.
About the Lagrange Linear PDE Calculator
A first-order linear PDE P p + Q q = R (p = ∂z/∂x, q = ∂z/∂y) is solved by finding two independent solutions u(x, y, z) = c₁ and v(x, y, z) = c₂ of the auxiliary equations dx/P = dy/Q = dz/R. The general solution is φ(u, v) = 0 for an arbitrary function φ.
This page handles the textbook families: P, Q, R each in one variable (integrate each fraction), the multiplier family xP = yQ = zR (like yz p + zx q = xy), and P : Q : R = x : y : z.
The formula
Pp + Qq = R: dx/P = dy/Q = dz/R, φ(u, v) = 0
Worked example
x²p + y²q = z²
- dx/x² = dy/y²: 1/y − 1/x = c₁; dy/y² = dz/z²: 1/z − 1/y = c₂
- φ(1/y − 1/x, 1/z − 1/y) = 0
Things to keep in mind
- Equations that need a multiplier other than x, y, z, or a combination like (dx + dy)/(P + Q), aren't detected; Charpit's method (non-linear equations) isn't covered.
Questions people ask
Why an arbitrary function?
Any function of u and v is constant along the same curves, so every φ(u, v) = 0 gives a solution surface.
What are p and q?
p = ∂z/∂x and q = ∂z/∂y, the usual notation in PDE chapters.