Binomial differential equation

In mathematics, the binomial differential equation is an ordinary differential equation containing one or more functions of one independent variable and the derivatives of those functions.

For example:

${\displaystyle \left(y'\right)^{m}=f(x,y),}$ when ${\displaystyle m}$ is a natural number (i.e., a positive integer), and ${\displaystyle f(x,y)}$ is a polynomial in two variables (i.e., a bivariate polynomial).

The Solution

Let ${\displaystyle P\left({x,y}\right)=\left({x+y}\right)^{k}=\sum \limits _{j=0}^{k}{\left({\begin{array}{*{20}c}k\\j\\\end{array}}\right)x^{j}y^{k-j}}}$ be a polynomial in two variables of order ${\displaystyle k}$; where ${\displaystyle k}$ is a positive integer. The binomial differential equation becomes ${\displaystyle \left({y'}\right)^{m}=\left({x+y}\right)^{k};}$ using the substitution ${\displaystyle v=x+y}$, we get that ${\displaystyle v'=1+y'}$, therefore ${\displaystyle \left({v'-1}\right)^{m}=v^{k}}$ or we can write ${\displaystyle v'=1+v^{\textstyle {k \over m}}}$, which is a separable ordinary differential equation, hence

${\displaystyle {\frac {dv}{dx}}=1+v^{\textstyle {k \over m}}\Rightarrow {\frac {dv}{1+v^{\textstyle {k \over m}}}}=dx\Rightarrow \int {\frac {dv}{1+v^{\textstyle {k \over m}}}}=x+C.}$

Special cases:

- If ${\displaystyle m=k}$, we have the differential equation ${\displaystyle v'-1=v}$ and the solution is ${\displaystyle y\left(x\right)=Ce^{x}-x-1}$, where ${\displaystyle C}$ is a constant.

- If ${\displaystyle m|k}$, i.e., ${\displaystyle m}$ divides ${\displaystyle k}$ so that there is a positive integer ${\displaystyle n}$ such that ${\displaystyle k=nm}$, then the solution has the form ${\displaystyle \int {\frac {dv}{1+v^{n}}}=x+C}$. From the tables book of Gradshteyn and Ryzhik we found that

${\displaystyle \int {\frac {dv}{1+v^{n}}}=\left\{{\begin{array}{l}-{\frac {2}{n}}\sum \limits _{i=0}^{{\textstyle {n \over 2}}-1}{P_{i}\cos \left({{\frac {2i+1}{n}}\pi }\right)}+{\frac {2}{n}}\sum \limits _{i=0}^{{\textstyle {n \over 2}}-1}{Q_{i}\sin \left({{\frac {2i+1}{n}}\pi }\right)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n:\,{\rm {even\,\,integer}}\\\\{\frac {1}{n}}\ln \left({1+v}\right)-{\frac {2}{n}}\sum \limits _{i=0}^{\textstyle {{n-3} \over 2}}{P_{i}\cos \left({{\frac {2i+1}{n}}\pi }\right)}+{\frac {2}{n}}\sum \limits _{i=0}^{\textstyle {{n-3} \over 2}}{Q_{i}\sin \left({{\frac {2i+1}{n}}\pi }\right)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,n:\,{\rm {odd\,\,integer}}\\\end{array}}\right.}$

and

${\displaystyle P_{i}={\frac {1}{2}}\ln \left({v^{2}-2v\cos \left({{\frac {2i+1}{n}}\pi }\right)+1}\right),}$

${\displaystyle Q_{i}=\arctan \left({\frac {v-\cos \left({{\textstyle {{2i+1} \over n}}\pi }\right)}{\sin \left({{\textstyle {{2i+1} \over n}}\pi }\right)}}\right).}$

See also

• Examples of differential equations

References

• Zwillinger, Daniel Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.