Chaplygin Inequality

one of the most important differential inequalities. Suppose y′(x) = f(x, y) and the functions u(x) and v(x) satisfy the differential inequalities

u′(x) – f(x, u) > 0

v′(x) – f(x, v) < 0

where x₀ ≤ x ≤ x₁. If u(x₀) = v(x₀) = y₀, then the solution y(x) of the differential equation y′(x) = f(x, y) that passes through the point (x₀, y₀) is contained between the functions u(x) and v(x); that is, u(x) > y(x) > v(x), where x₀ < x ≤ x₁. This theorem, of which the simplest case is presented here, was proved by S. A. Chaplygin in 1919 and was used by him as the basis of a method for the approximate integration of differential equations. Chaplygin proved a similar theorem for the equation y⁽ⁿ⁾ – f(x, y, y′, . . . y^(n–1) = 0 and extended the theorem to partial differential equations.