Binomial differential equation

In mathematics, the binomial differential equation is an ordinary differential equation of the form ${\displaystyle \left(y'\right)^{m}=f(x,y),}$ where ${\displaystyle m}$ is a natural number and ${\displaystyle f(x,y)}$ is a polynomial that is analytic in both variables.^[1][2]

Let ${\displaystyle P(x,y)=(x+y)^{k}}$ be a polynomial of two variables of order ${\displaystyle k}$, where ${\displaystyle k}$ is a natural number. By the binomial formula,

${\displaystyle P(x,y)=\sum \limits _{j=0}^{k}{{\binom {k}{j}}x^{j}y^{k-j}}}$.

The binomial differential equation becomes ${\textstyle (y')^{m}=(x+y)^{k}}$. Substituting ${\displaystyle v=x+y}$ and its derivative ${\displaystyle v'=1+y'}$ gives ${\textstyle (v'-1)^{m}=v^{k}}$, which can be written ${\textstyle {\tfrac {dv}{dx}}=1+v^{\tfrac {k}{m}}}$, which is a separable ordinary differential equation. Solving gives

${\displaystyle {\begin{array}{lrl}&{\frac {dv}{dx}}&=1+v^{\tfrac {k}{m}}\\\Rightarrow &{\frac {dv}{1+v^{\tfrac {k}{m}}}}&=dx\\\Rightarrow &\int {\frac {dv}{1+v^{\tfrac {k}{m}}}}&=x+C\end{array}}}$

• If ${\displaystyle m=k}$, this gives 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 ${\displaystyle m|k}$ (that is, ${\displaystyle m}$ is a divisor of ${\displaystyle k}$), then the solution has the form ${\textstyle \int {\frac {dv}{1+v^{n}}}=x+C}$. In the tables book Gradshteyn and Ryzhik, this form decomposes as:

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

where

${\displaystyle {\begin{aligned}P_{i}&={\frac {1}{2}}\ln \left({v^{2}-2v\cos \left({{\frac {2i+1}{n}}\pi }\right)+1}\right)\\Q_{i}&=\arctan \left({\frac {v-\cos \left({{\textstyle {{2i+1} \over n}}\pi }\right)}{\sin \left({{\textstyle {{2i+1} \over n}}\pi }\right)}}\right)\end{aligned}}}$

• Examples of differential equations