Binomial differential equation

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

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

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

• Examples of differential equations

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