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N - r. But these general solutions are not essentially different. For the partial equation of the second order F(x, y, z, p, 4, r, s, t) = 0, this deli- We can regard equation (8) as a relation between the point (x, y, z) on the integral surface S and the direction cosines of a tangent plane at that point. Therefore the tangent planes at all points of the surface form a one-parameter family. They envelop a cone (T) whose vertex is (x, y, z) on S. The tangent plane at a point M on the integral surface S is tangent to this cone (T) along one generating line G of (T).

Example 2. If PI, P2, . . , P,,, R are functions of independent variables x1, . . > =o G’ax,ax,‘“’ > that involves a function z of independent variables x1, x2, . . x,, its tpartial derivatives, and the independent variables x1, . ,x,. The definition of a system of partial differential equations is similar to that of a system of tordinary differential equations. ) The torder of the highest derivative appearing in a partial differential equation is called the order of the partial differential equation.

T. Schwartz, Linear operators II, Interscience, 1963. [6] B. M. Levitan and I. S. Sargsyan (Sargsjan), Introduction to spectral theory: Self-adjoint ordinary differential operators, Aml:r. Math. Sot. Transl. of Math. Monographs, 1975. 3) Ordinary Differential Equations (Initial Value Problems) A. General Consider equations Remarks a system of ordinary dy,ldx=fi(x,~,, . ,n. (1) A. L. , n, and their derivatives aJ/ay, are continuous in a neighborhood Iof a point (a, b,, . . , n. These conditions are called initial conditions, and the values a, b,, .

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Encyclopedia Dictionary of Mathematics (Section C)

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