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

Hence the envelope has equation. These solutions correspond to spheres whose radius grows or shrinks with velocity c. These are light cones in space-time. The solution is obtained by taking the envelope of all the spheres with centers on S , whose radii grow with velocity c. This envelope is obtained by requiring that. Thus the envelope corresponds to motion with velocity c along each normal to S.

## Handbook of First-Order Partial Differential Equations

The normals to S are the light rays. The notation is relatively simple in two space dimensions, but the main ideas generalize to higher dimensions. A general first-order partial differential equation has the form. There are n parameters required in the n -dimensional case. Each choice of the function w leads to a solution of the PDE.