# Complete vs General Integral of first order PDE

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?

Thanks.

-

The notion "complete integral" here refers to solutions of specific (1st order) PDEs that depend on the maximal number of constants of motion. If you want a concrete example I can refer you to Equation (10) of this paper, or even better to Ref. [10] in that paper.

A "general solution", by contrast, need not depend explicitly on constants of motion, but usually contains some free (integration) function. As an example for a solution not depending explicitly on constants of motion see the "enveloping solution" in Eq. (11) of the paper above.

[I never encountered these notions anywhere, except when solving Hamilton-Jacobi equations - this seems to be also the context to which Landau refers.]

-
Landau does refer to Hamilton-Jacobi equations. Can you point me to some other references where the same notions are used relating to Hamilton-Jacobi equations (in classical mechanics)? Perhaps seeing them will clarify me. –  becko Aug 18 '11 at 3:08
I can refer you to the book mentioned in the paper I referred to in my reply. It is, however, in German: E. Kamke, Differentialgleichungen, Vol II. But I am sure there is some English book dealing with this. You might try e.g. A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. The notion of "complete integral" and "general solution" is also defined on this useful webpage: eqworld.ipmnet.ru/en/solutions/fpde/fpdetoc3.htm –  Daniel Grumiller Aug 18 '11 at 5:52

In Russian "integral" is a synonym of a solution of differential equation. "general integral" means general solution, "complete" probably means sum of particular solution and general solution (called the complementary solution)

iPDEs

-
This answer is very confusing. Particular solution usually means a solution to an inhomogeneous equation while what you and your reference mean is the same general solution, only with some boundary/initial conditions plugged-in. –  Marek Aug 16 '11 at 6:19
thx, I corrected –  troyaner Aug 16 '11 at 7:51