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. Characteristics of first-order partial differential equation 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 simplicity, 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 Suppose that a solution z is known, and consider the surface graph z = z(x,y) in R3. A normal vector to this surface is given by As a result, equation () is equivalent to the geometrical statement that the vector field is tangent to the surface z = z(x,y) at every point, for the dot product of this vector field with the above normal vector is zero. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation and are given by the Lagrange–Charpit equations A parametrization invariant form of the Lagrange–Charpit equations is: Linear and quasilinear cases Consider now a PDE of the form For this PDE to be linear, the coefficients ai may be functions of the spatial variables only, and independent of u. For it to be quasilinear, ai may also depend on the value of the function, but not on any derivatives. The distinction between these two cases is inessential for the discussion here. For a linear or quasilinear PDE, the characteristic curves are given parametrically by such that the following system of ODEs is satisfied Equations () and () give the characteristics of the PDE. Proof for quasilinear Case In the quasilinear case, the use of the method of characteristics is justified by Grönwall's inequality. The above equation may be written as We must distinguish between the