A Darboux differential equation is a first-order ordinary differential equation of the form
 |
(1)
|
where 
is the derivative of 
, 
and 
are homogeneous polynomials
of the same degree 
, and 
is a homogeneous polynomial
of degree 
or is zero (in which case 
can be set to 
).
For 
, substituting 
and writing 
, 
, and 
gives
![u^'=(p(u)-uq(u))/(x[q(u)+x^(m-n+1)r(u)]).](/images/equations/DarbouxDifferentialEquation/NumberedEquation2.svg) |
(2)
|
Here 
. Where the denominators
are nonzero and 
,
inversion gives
 |
(3)
|
The inverse equation is a Bernoulli differential equation if the exponent 
is neither 0 nor 1, and a linear equation otherwise (Polyanin
and Zaitsev 2003). If 
or 
,
the original equation is homogeneous in 
and 
.
A zero 
of 
also gives the straight-line solution 
wherever the original denominator is nonzero.
