Binomial differential equation

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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.

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

Solution

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

[math]\displaystyle{ P(x,y) = \sum\limits_{j = 0}^k { \binom{k}{j} x^j y^{k - j} } }[/math].Template:Relevant?

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

[math]\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} }[/math]

Special cases

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

[math]\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. }[/math]

where

[math]\displaystyle{ \begin{align} 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{align} }[/math]

See also

• Examples of differential equations