Solution of Second Order Differential Equation with Missing Independent Variable
Theorem
Let $\map g {y, \dfrac {\d y} {\d x}, \dfrac {\d^2 y} {\d x^2} } = 0$ 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:
- $(1): \quad \map g {y, \dfrac {\d y} {\d x}, \dfrac {\d^2 y} {\d x^2} } = 0$
Let a new dependent variable $p$ be introduced:
- $y' = p$
Hence:
- $y'' = \dfrac {\d p} {\d x} = \dfrac {\d p} {\d y} \dfrac {\d y} {\d x} = p \dfrac {\d p} {\d y}$
Then $(1)$ can be transformed into:
- $(2): \quad \map g {y, p, p \dfrac {\d p} {\d y} = 0}$
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:
- $\map g {x, \dfrac {\d y} {\d x} }$
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.
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.11$: Reduction of Order