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.