First Integral

The first integral of a system of ordinary differential equations

is a relation of the form

ϕ (x, y₁, …, y_n) = C

where C is an arbitrary constant. The left side of the relation remains constant when any solution y₁ = y₁(x), …, y_n = y_n(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 … y_n, 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 ϕ_i (x, y₁ …, y_n) = C_i (i = 1, …, k; k < n) of a system are

Figure 1

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.