Solution of Second Order Differential Equation with Missing Dependent Variable

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Theorem

Let $\map f {x, y', y''} = 0$ 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:

$(1): \quad \map f {x, y', y''} = 0$

Let a new dependent variable $p$ be introduced:

$y' = p$
$y'' = \dfrac {\d p} {\d x}$

Then $(1)$ can be transformed into:

$(2): \quad \map f {x, p, \dfrac {\d p} {\d x} } = 0$

which is a first order ODE.


If $(2)$ has a solution which can readily be found, it will be expressible in the form:

$(3): \quad \map g {x, p}$

which can then be expressed in the form:

$\map g {x, \dfrac {\d y} {\d x} } = 0$

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.

$\blacksquare$


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