Английская Википедия:Binomial differential equation
Шаблон:Multiple issues 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:Шаблон:Clarify
- <math>\left( y' \right)^m = f(x,y),</math> when <math>m</math> is a natural number and <math>f(x,y)</math> is a polynomial of two variables (bivariate).
Solution
Let <math> P(x,y) = (x + y)^k </math> be a polynomial of two variables of order <math>k</math>, where <math>k</math> is a natural number. By the binomial formula,
- <math> P(x,y) = \sum\limits_{j = 0}^k { \binom{k}{j} x^j y^{k - j} } </math>.Шаблон:Relevant?
The binomial differential equation becomes <math display="inline">(y')^m = (x + y)^k</math>.Шаблон:Unclear inline Substituting <math>v = x + y</math> and its derivative <math>v' = 1 + y'</math> gives <math display="inline">(v'-1)^m = v^k</math>, which can be written <math display="inline">\tfrac{dv}{dx} = 1 + v^{\tfrac{k}{m}}</math>, which is a separable ordinary differential equation. Solving gives
- <math>
\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>m=k</math>, this gives the differential equation <math>v' - 1 = v</math> and the solution is <math>y\left( x \right) = Ce^x - x - 1</math>, where <math>C</math> is a constant.
- If <math>m|k</math> (that is, <math>m</math> is a divisor of <math>k</math>), then the solution has the form <math display="inline">\int {\fracШаблон:DvШаблон:1 + v^n} = x + C</math>. In the tables book Gradshteyn and Ryzhik, this form decomposes as:
- <math>
\int {\fracШаблон:DvШаблон:1 + v^n} = \left\{ \begin{array}{ll}
- \frac{2}{n}\sum\limits_{i = 0}^{{\textstyle{n \over 2}} - 1} {P_i \cos \left( {\fracШаблон: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>
\begin{align} P_i &= \frac{1}{2}\ln \left( {v^2 - 2v\cos \left( {\fracШаблон: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
References
