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The following is an excerpt from Landau's Course on Theoretical Physics Vol.1 Mechanics:

... we should recall the fact that every first-order partial differential equation has a solution depending on an arbitrary function; such a solution is called the general integral of the equation. In mechanical applications, the general integral of the Hamilton-Jacobi equation is less important than a complete integral, which contains as many independent arbitrary constants as there are independent variables.

Can someone clarify what's a complete integral and what's a general integral of a first order partial differential equation?


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A 1st order PDE (for a function of $n$ variables) is equivalent to an ODE in dimension $2n+1$ (its integral curves are called characteristics). A complete integral is a family of solutions of the PDE which is large enough, so that we can determine characteristics out of it, and then we can use those characteristics to find all solutions of the PDE. – user8268 Apr 26 '11 at 6:32

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