Binomial differential equation explained

In mathematics, the binomial differential equation is an ordinary differential equation of the form

\left( y' \right) ^{m} = f(x,y), where m is a natural number and f(x,y) is a polynomial that is analytic in both variables.^{[1] [2]}

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 ^{k} _{j =0} { \binom{k}{j} x ^{j} y ^{k} } .The binomial differential equation becomes $(y\text{'})^m\; =\; (x\; +\; y)^k$. Substituting v = x + y and its derivative v' = 1 + y' gives $(v\text{'}-1)^m\; =\; v^k$, which can be written $\backslash tfrac\; =\; 1\; +\; v^$, which is a separable ordinary differential equation. Solving gives\begin{array}{lrl} & (dv)/(dx) & = 1 + v ^{\tfrac{k} {m}} \ ⇒ & (dv)/(1+ v ^{\tfrac{k} {m) }} &= dx \\ \Rightarrow & \int &= x + C \end

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 $\backslash int\; =\; x\; +\; C$. In the tables book Gradshteyn and Ryzhik, this form decomposes as:\int { ({dv)/() }} = \left\

Notes and References

1. Book: Hille, Einar . Lectures on ordinary differential equations . 1894 . . 978-0201530834 . 675.
2. Book: Zwillinger, Daniel . Handbook of differential equations . 1998 . Academic Press . 978-0-12-784396-4 . 3rd . San Diego, Calif . 180.