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.

REFERENCE

Stepanov, V. V. Kurs differentsial’nykh uravnenii, 8th ed. Moscow, 1959.

单词	first integral
释义	First Integral 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. REFERENCE Stepanov, V. V. Kurs differentsial’nykh uravnenii, 8th ed. Moscow, 1959.
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