Method of characteristics

In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface.

For a first-order PDE (partial differential equation), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.

For the sake of motivation, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear PDE of the form

${\displaystyle a(x,y,z){\frac {\partial z}{\partial x}}+b(x,y,z){\frac {\partial z}{\partial y}}=c(x,y,z).}$

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