Rearrangement of Variables in Total Differential Equation

Theorem

In a total differential equation, any one of the variables can be regarded as an independent variable, while the remainder can be treated as dependent variables.

Proof by Example

Consider the total differential equation:

$u \rd x + v \rd y + w \rd z = 0$

This may be written:

$u + v \dfrac {\d y} {\d x} + w \dfrac {\d z} {\d x} = 0$

or another independent variable $t$ may be introduced:

$u \dfrac {\d x} {\d t} + v \dfrac {\d y} {\d t} + w \dfrac {\d z} {\d t} = 0$

Sources

• 1926: E.L. Ince: Ordinary Differential Equations ... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.1$ Definitions