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General Properties of the Differential Equations

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Part of the Astrophysics and Space Science Library book series (ASSL,volume 443)

Abstract

Consider a system of differential equations:

$$ \frac{{{\text{d}}x_{1} }}{{{\text{d}}t}} = X_{1} ,\quad \frac{{{\text{d}}x_{2} }}{{{\text{d}}t}} = X_{2} ,\ldots \frac{{{\text{d}}x_{n} }}{{{\text{d}}t}} = X_{n} , $$
(1)

where \( t \) represents the independent variable that we will call time and \( x_{1} ,x_{2} ,\; \ldots x_{n} \) the unknown functions, where \( X_{1} ,X_{2} , \ldots \,X_{n} \) are given functions of \( x_{1} ,x_{2} , \ldots \,x_{n} \). We suppose in general that the functions \( X_{1} ,X_{2} , \ldots \,X_{n} \) are analytic and one-to-one for all real values of \( x_{1} ,x_{2} , \ldots \,x_{n} \).

Keywords

  • Singular Point
  • Constant Coefficient
  • Convex Polygon
  • Angular Variable
  • Specific Solution

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    In my thesis, I did not state this restriction and I did not assume that the sum of the \( m \) was larger than 1. It would therefore seem that the theorem is incorrect when for example \( \lambda_{2} = \lambda_{1} \). This is not the case. If we had

    $$ m_{2} \lambda_{2} + \cdots m_{n} \lambda_{n} = \lambda_{1} {\text{for }}\left( {m_{2} + m_{3} + \cdots m_{n} > 1} \right) $$

    some coefficients of the expansion would take the form \( A/0 \) and would become infinite. This is the reason why we had to assume that such a relationship does not hold. If on the other hand \( \lambda_{2} = \lambda_{1} \), then some coefficients would take the form \( 0/0 \).

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Correspondence to Henri Poincaré .

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Poincaré, H. (2017). General Properties of the Differential Equations. In: The Three-Body Problem and the Equations of Dynamics. Astrophysics and Space Science Library, vol 443. Springer, Cham. https://doi.org/10.1007/978-3-319-52899-1_1

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