In mathematics, the binomial differential equation is an ordinary differential equation of the form where is a natural number and is a polynomial that is analytic in both variables.
Solution Let be a polynomial of two variables of order , where is a natural number. By the binomial formula, . The binomial differential equation becomes . Substituting and its derivative gives , which can be written , which is a separable ordinary differential equation. Solving gives
Special cases
If , this gives the differential equation and the solution is , where is a constant. If (that is, is a divisor of ), then the solution has the form . In the tables book Gradshteyn and Ryzhik, this form decomposes as:
where
See also Examples of differential equations
References