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.