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
![{\displaystyle {\begin{array}{lrl}&{\frac {dv}{dx}}&=1+v^{\tfrac {k}{m}}\\\Rightarrow &{\frac {dv}{1+v^{\tfrac {k}{m}}}}&=dx\\\Rightarrow &\int {\frac {dv}{1+v^{\tfrac {k}{m}}}}&=x+C\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04cdb46f595f97d96d782af4ab45d5194ed44d32)
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:
![{\displaystyle \int {\frac {dv}{1+v^{n}}}=\left\{{\begin{array}{ll}-{\frac {2}{n}}\sum \limits _{i=0}^{{\textstyle {n \over 2}}-1}{P_{i}\cos \left({{\frac {2i+1}{n}}\pi }\right)}+{\frac {2}{n}}\sum \limits _{i=0}^{{\tfrac {n}{2}}-1}{Q_{i}\sin \left({{\frac {2i+1}{n}}\pi }\right)},&n:{\text{even integer}}\\\\{\frac {1}{n}}\ln \left({1+v}\right)-{\frac {2}{n}}\sum \limits _{i=0}^{\textstyle {{n-3} \over 2}}{P_{i}\cos \left({{\frac {2i+1}{n}}\pi }\right)}+{\frac {2}{n}}\sum \limits _{i=0}^{\tfrac {n-3}{2}}{Q_{i}\sin \left({{\frac {2i+1}{n}}\pi }\right)},&n:{\text{odd integer}}\\\end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e42e3960a9091623a9e4e2839fbd5e85dc079a0)
where
![{\displaystyle {\begin{aligned}P_{i}&={\frac {1}{2}}\ln \left({v^{2}-2v\cos \left({{\frac {2i+1}{n}}\pi }\right)+1}\right)\\Q_{i}&=\arctan \left({\frac {v-\cos \left({{\textstyle {{2i+1} \over n}}\pi }\right)}{\sin \left({{\textstyle {{2i+1} \over n}}\pi }\right)}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57f414d2049c9b3af3fec7100a6118f9d05803e0)
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.