Solution of Second Order Differential Equation with Missing Dependent Variable

Theorem
Let $f \left({x, y', y''}\right) = 0$ be a second order ordinary differential equation in which the dependent variable $y$ is not explicitly present.

Then $f$ can be reduced to a first order ordinary differential equation, whose solution can be determined.

Proof
Consider the second order ordinary differential equation:
 * $(1): \quad f \left({x, y', y''}\right) = 0$

Let a new dependent variable $p$ be introduced:
 * $y' = p$
 * $y'' = \dfrac {\mathrm d p} {\mathrm d x}$

Then $(1)$ can be transformed into:
 * $(2): \quad f \left({x, p, \dfrac {\mathrm d p} {\mathrm d x} }\right) = 0$

which is a first order ODE.

If $(2)$ has a solution which can readily be found, it will be expressible in the form:
 * $(3): \quad g \left({x, p}\right)$

which can then be expressed in the form:
 * $g \left({x, \dfrac {\mathrm d y} {\mathrm d x}}\right) = 0$

which is likewise subject to the techniques of solution of a first order ODE.

Hence such a second order ODE is reduced to the problem of solving two first order ODEs in succession.