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general solution[¦jen·rəl sə′lü·shən]
A general solution of a differential equation
y(n) = f(x, y, y’, …, y(n–1))
is a family of functions
y = ø(x, C1, …, Cn)
that are continuously dependent on n arbitrary constants C1, …, Cn such that when these constants are appropriately selected, we can obtain any solution of the equation (a particular solution) unambiguously defined by the initial data, which fill a certain region of n-dimensional space. (SeeDIFFERENTIAL EQUATIONS.) If every function y defined by the relation ø(x, y, C1, …, Cn) = 0 and satisfying the corresponding smoothness conditions is a general solution of the differential equation, then this relation is called a general integral of the differential equation. For example, for the differential equation y’ = –x/y, the functions (upper semicircles) and y (lower semicircles) are general solutions; the relation x2 + y2 = C2 (a family of circles) is a general integral (see Figure 1).
A general solution is similarly defined for a system of ordinary differential equations.