ABSTRACT
Such equations are encountered in analytical mechanics, calculus of variations, optimal control, differential games, dynamic programming, geometric optics, differential geometry, and other fields.
In this subsection, we consider only smooth solutions w =w(x, y) of equation (24.1.1.1), which are continuously differentiable with respect to both arguments (Subsection 24.3 deals with nonsmooth solutions). 1◦. Let a particular solution of equation (24.1.1.1),
w = Ξ(x, y,C1,C2), (24.1.1.2)
depending on two parameters C1 and C2, be known. The two-parameter family of solutions (24.1.1.2) is called a complete integral of equation (24.1.1.1) if the rank of the matrix
M =
( Ξ1 Ξx1 Ξy1 Ξ2 Ξx2 Ξy2
) (24.1.1.3)
is equal to two in the domain being considered (for example, this is valid if Ξx1Ξy2 – Ξx2Ξy1 ≠ 0). In equation (24.1.1.3), Ξn denotes the partial derivative of Ξ with respect to Cn (n = 1, 2), Ξxn is the second partial derivative with respect to x and Cn, and Ξyn is the second partial derivative with respect to y and Cn.
