Definition:Homogeneous Differential Equation/Also presented as

From ProofWiki
Jump to navigation Jump to search

Homogeneous Differential Equation: Also presented as

A homogeneous differential equation can also be presented as:

$\dfrac {\d y} {\d x} + \dfrac {\map M {x, y} } {\map N {x, y} } = 0$

or:

$\dfrac {\d y} {\d x} = \dfrac {\map M {x, y} } {\map N {x, y} }$

where both $M$ and $N$ are homogeneous functions of the same degree.


However, note that in the latter case the sign has changed, therefore care needs to be taken when applying the formula.


Some sources present this using the language of differentials in the form:

$\map M {x, y} \rd x = \map N {x, y} \rd y$

or:

$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$


Some sources jump straight to the form:

$\dfrac {\d y} {\d x} = \map F {\dfrac y x}$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Homogeneous_Differential_Equation/Also_presented_as&oldid=686269"