---
tag: H90CS9B8YH
kind: concept
name: Complete integral
msc: [35F20, 35A24]
sources:
  - url: https://en.wikipedia.org/wiki/First-order_partial_differential_equation
---

## Summary

A complete integral of a first-order [[partial differential equation|GC5BDZ6QBC]] in $n$ variables is a family of solutions depending on $n$ independent parameters.
From such a family one builds many more solutions by taking envelopes of subfamilies.
Complete integrals are the classical route to solving [[Hamilton–Jacobi equations|5TRXQKP7DY]] in mechanics and Clairaut-type equations.
See the [[definition|KEC5TGS9WM]].

## Explanations

### L1 human

For a first-order equation in $n$ variables, one often finds a simple family of solutions, such as planes with a free slope.
If the family has $n$ genuinely different parameters, it is called a complete integral.
For example, every plane $u = a_1 x_1 + a_2 x_2 + a_1^2 + a_2^2$ solves the equation $u = x \cdot \nabla u + \abs{\nabla u}^2$.
Bending these planes together, by taking the envelope of a one-parameter subfamily, produces new curved solutions.