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Binomial differential equation

From Wikipedia, the free encyclopedia

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

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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

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  • 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

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References

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  1. ^ Hille, Einar (1894). Lectures on ordinary differential equations. Addison-Wesley Publishing Company. p. 675. ISBN 978-0201530834.
  2. ^ Zwillinger, Daniel (1998). Handbook of differential equations (3rd ed.). San Diego, Calif: Academic Press. p. 180. ISBN 978-0-12-784396-4.