The first integral of a system of ordinary differential equations
is a relation of the form
ϕ (x, y1, …, yn) = C
where C is an arbitrary constant. The left side of the relation remains constant when any solution y1 = y1(x), …, yn = yn(x) of the system is substituted but is not a fixed constant. Geometrically, the first integral is a family of hypersurfaces in the (n + 1)-dimensional space Oxy1 … yn, in each of which there exists a subfamily of the integral curves of the system. For example, y2 + z2 = C2 (circular cylinder) is a first integral of the system dy/dx = z, dz/dx = – y; the integral curves y = C sin (x – x0) and z = C cos (x – x0) are helices on the cylinders (see Figure 1). If k independent first integrals ϕi (x, y1 …, yn) = Ci (i = 1, …, k; k < n) of a system are
known, the system’s order can in general be decreased by k units. If k = n, the general integral of the system is obtained without integration.