First-order partial differential equation

In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form

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. 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.

Characteristic surfaces for the wave equation are level surfaces for solutions of the equation

There is little loss of generality if we set ${\displaystyle u_{t}=1}$: in that case u satisfies

In vector notation, let

A family of solutions with planes as level surfaces is given by

where

If x and x₀ are held fixed, the envelope of these solutions is obtained by finding a point on the sphere of radius 1/c where the value of u is stationary. This is true if ${\displaystyle {\vec {p}}}$ is parallel to ${\displaystyle {\vec {x}}-{\vec {x_{0}}}}$. Hence the envelope has equation

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