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**Jose27** Basically a characteristic is a curve along which the directional derivative is known and given by the coefficients. To get these we solve an ODE, and if we are given initial data (Cauchy problem) we have initial conditions to impose, so everything is determined. On the other hand if you want the general solution this initial conditions are missing so you get a family of curves all satisfying the same differential equation, and if this curve intersects a solution surface, by fixing this point and using the existence and uniqueness theorem for ODE, we get that there is an 'interval' around this point in the curve which is contained in the surface; we can then conclude that the whole characteristic is contained in the surface as long as both exist.