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:

$\left(y'\right)^{m}=f(x,y),$ when $m$ is a natural number and $f(x,y)$ is a polynomial of two variables (bivariate).

Solution

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

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

The binomial differential equation becomes ${\textstyle (y')^{m}=(x+y)^{k}}$. Substituting $v=x+y$ and its derivative $v'=1+y'$ 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

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

Special cases

• If $m=k$, this gives the differential equation $v'-1=v$ and the solution is $y\left(x\right)=Ce^{x}-x-1$, where $C$ is a constant.
• If $m|k$ (that is, $m$ is a divisor of $k$), 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:

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

where

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

See also

• Examples of differential equations

References

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