solutions to the ODE and the solutions to the PDE, which we do not know are equal a priori. Letting capital letters be the solutions to the ODE we find Examining , we find, upon differentiating that which is the same as We cannot conclude the above is 0 as we would like, since the PDE only guarantees us that this relationship is satisfied for , , and we do not yet know that . However, we can see that since by the PDE, the last term is 0. This equals By the triangle inequality, we have Assuming are at least , we can bound this for small times. Choose a neighborhood around small enough such that are locally Lipschitz. By continuity, will remain in for small enough . Since , we also have that will be in for small enough by continuity. So, and for . Additionally, for some for by compactness. From this, we find the above is bounded as for some . It is a straightforward application of Grönwall's Inequality to show that since we have for as long as this inequality holds. We have some interval such that in this interval. Choose the largest such that this is true. Then, by continuity, . Provided the ODE still has a solution in some interval after , we can repeat the argument above to find that in a larger interval. Thus, so long as the ODE has a solution, we have . Fully nonlinear case Consider the partial differential equation where the variables pi are shorthand for the partial derivatives Let (xi(s),u(s),pi(s)) be a curve in R2n+1. Suppose that u is any solution, and that Along a solution, differentiating () with respect to s gives The second equation follows from applying the chain rule to a solution u, and the third follows by taking an exterior derivative of the relation . Manipulating these equations gives where λ is a constant. Writing these equations more symmetrically, one obtains the Lagrange–Charpit equations for the characteristic Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the Monge cone of the differential equation should everywhere be tangent to the graph of the solution. The second order partial differential equation is solved with Charpit method . Example As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs).