Editing Separation of variables (section)

=== Generalization of separable ODEs to the nth order ===
Much like one can speak of a separable first-order ODE, one can speak of a separable second-order, third-order or ''n''th-order ODE. Consider the separable first-order ODE:
:<math>\frac{dy}{dx}=f(y)g(x)</math>
The derivative can alternatively be written the following way to underscore that it is an operator working on the unknown function, ''y'':
:<math>\frac{dy}{dx}=\frac{d}{dx}(y)</math>
Thus, when one separates variables for first-order equations, one in fact moves the ''dx'' denominator of the operator to the side with the ''x'' variable, and the ''d''(''y'') is left on the side with the ''y'' variable. The second-derivative operator, by analogy, breaks down as follows:
:<math>\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dx}\left(\frac{d}{dx}(y)\right)</math>
The third-, fourth- and ''n''th-derivative operators break down in the same way. Thus, much like a first-order separable ODE is reducible to the form
:<math>\frac{dy}{dx}=f(y)g(x)</math>
a separable second-order ODE is reducible to the form
:<math>\frac{d^2y}{dx^2}=f\left(y'\right)g(x)</math>
and an nth-order separable ODE is reducible to
:<math>\frac{d^ny}{dx^n}=f\!\left(y^{(n-1)}\right)g(x)</math>