Solution of Second Order Differential Equation with Missing Independent Variable

Theorem
Let $g \left({y, \dfrac {\mathrm d y} {\mathrm d x}, \dfrac {\mathrm d^2 y} {\mathrm d x^2} }\right) = 0$ be a second order ordinary differential equation in which the independent variable $x$ is not explicitly present.

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

Proof
Consider the second order ordinary differential equation:
 * $(1): \quad g \left({y, \dfrac {\mathrm d y} {\mathrm d x}, \dfrac {\mathrm d^2 y} {\mathrm d x^2} }\right) = 0$

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

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

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:
 * $g \left({x, \dfrac {\mathrm d y} {\mathrm d x}}\right)$

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.