59 - Exact Second Order Equations

Abstract

This chapter discusses solution of exact second-order equations. It is applicable to some nonlinear second-order ordinary differential equations of the form f(x,y,y')y“ + g(x, y, y') = 0. The solution yields a first integral (which will be a first-order ordinary differential equation). The second-order differential equation F(x, y, y', y“) = 0 is said to be exact if it is the total differential of some function; that is, F = dф/dx, where ф = ф(x, y, y'). If F(x, y, y', y“) is exact, then ф = C is a solution to F(x, y, y', y“), with C an arbitrary constant. Differentiating ф= C with respect to x gives dф/dx = ∂ф/∂x + ∂ф/∂y(y') + ∂ф/∂y'(y“). The chapter also highlights that exact second-order linear ordinary differential equations have factorable operators.