First-order partial differential equation

In mathematics, a first-order partial differential equation is a partial differential equation that involves the first derivatives of an unknown function ${\displaystyle u}$ of ${\displaystyle n\geq 2}$ variables. The equation takes the form^[1] $${\displaystyle F(x_{1},\ldots ,x_{n},u,u_{x_{1}},\ldots u_{x_{n}})=0,}$$ using subscript notation to denote the partial derivatives of ${\displaystyle u}$.

Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics, e.g., the advection equation. If a family of solutions of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.