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 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 solutionsof 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. The general solution to the first order partial differential equation is a solution which contains an arbitrary function. But, the solution to the first order partial differential equations with as many arbitrary constants as the number of independent variables is called the complete integral. The following n-parameter family of solutions is a complete integral if det | ϕ x i a j | ≠ 0 {displaystyle { ext{det}}|phi _{x_{i}a_{j}}| eq 0} . Characteristic surfaces for the wave equation are level surfaces for solutions of the equation There is little loss of generality if we set u t = 1 {displaystyle u_{t}=1} : in that case u satisfies