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.[1][2]
Solution
Let be a polynomial of two variables of order , where is a natural number. By the binomial formula,
- .[relevant?]
The binomial differential equation becomes .[clarification needed] 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
References
- ^ Hille, Einar (1894). Lectures on ordinary differential equations. Addison-Wesley Publishing Company. p. 675. ISBN 978-0201530834.
- ^ Zwillinger, Daniel (1998). Handbook of differential equations (3rd ed.). San Diego, Calif: Academic Press. p. 180. ISBN 978-0-12-784396-4.