Binomial differential equation

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

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,

$P(x,y)=\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})x^j{y}^{k-j}$.Template:Relevant?

The binomial differential equation becomes $(y^{\prime }{)}^m=(x+y{)}^k$.Template:Unclear inline Substituting $v=x+y$ and its derivative $v^{\prime }=1+y^{\prime }$ gives $(v^{\prime }-1{)}^m=v^k$, which can be written $\frac{dv}{dx}=1+v^{\frac{k}{m}}$, which is a separable ordinary differential equation. Solving gives

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

Special cases

• If $m=k$, this gives the differential equation $v^{\prime }-1=v$ and the solution is $y\left(x\right)=C{e}^x-x-1$, where $C$ is a constant.
• If $m|k$ (that is, $m$ is a divisor of $k$), then the solution has the form $\int \frac{dv}{1+v^n}=x+C$. In the tables book Gradshteyn and Ryzhik, this form decomposes as:

$\int \frac{dv}{1+v^n}=\{\begin{array}{cc}-\frac{2}{n}\sum _{i=0}^{\frac{n}{2}-1}{P}_i\cos \left(\frac{2i+1}{n}\pi \right)+\frac{2}{n}\sum _{i=0}^{\frac{n}{2}-1}{Q}_i\sin \left(\frac{2i+1}{n}\pi \right),& n:\text{even integer}\\ \\ \frac{1}{n}\ln \left(1+v\right)-\frac{2}{n}\sum _{i=0}^{\frac{n-3}{2}}{P}_i\cos \left(\frac{2i+1}{n}\pi \right)+\frac{2}{n}\sum _{i=0}^{\frac{n-3}{2}}{Q}_i\sin \left(\frac{2i+1}{n}\pi \right),& n:\text{odd integer}\end{array}$

where

$\begin{array}{cc}{P}_i& =\frac{1}{2}\ln \left(v^2-2v\cos \left(\frac{2i+1}{n}\pi \right)+1\right)\\ Q_i& =\arctan \left(\frac{v-\cos \left(\frac{2i+1}{n}\pi \right)}{\sin \left(\frac{2i+1}{n}\pi \right)}\right)\end{array}$

See also

• Examples of differential equations