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In mathematics, the binomial differential equation is an ordinary differential equation containing one or more functions of one independent variable and the derivatives of those functions.
For example:
when
is a natural number (i.e., a positive integer), and
is a polynomial in two variables (i.e., a bivariate polynomial).
The Solution[]
Let
be a polynomial in two variables of order
; where
is a positive integer. The binomial differential equation becomes
using the substitution
, we get that
, therefore
or we can write
, which is a separable ordinary differential equation, hence
Special cases:
- If
, we have the differential equation
and the solution is
, where
is a constant.
- If
, i.e.,
divides
so that there is a positive integer
such that
, then the solution has the form
. From the tables book of Gradshteyn and Ryzhik we found that
and
See also[]
References[]
- Zwillinger, Daniel Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.