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词条 Binomial differential equation
释义

  1. The Solution

  2. References

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The binomial differential equation is the ordinary differential equation

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

References

  • Zwillinger, Daniel Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.
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2 : Differential calculus|Ordinary differential equations
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