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Revision difference : solutions of ordinary differential equation |
| Version current |
Version 1 |
| Let us consider the ordinary differential equation |
Let us consider the ordinary differential equation |
| \begin{align} |
\begin{align} |
| F(x,\,y,\,y',\,y'',\,\ldots,\,y^{(n)}) = 0 |
F(x,\,y,\,y',\,y'',\,\ldots,\,y^{(n)}) = 0 |
| \end{align} |
\end{align} |
| of order $n$. |
of order $n$. |
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| The {\em general solution} of (1) is a function |
The {\em general solution} of (1) is a function |
| $$x\mapsto y = \varphi(x,\,C_1,\,C_2,\,\ldots,\,C_n)$$ |
$$x\mapsto y = \varphi(x,\,C_1,\,C_2,\,\ldots,\,C_n)$$ |
| satisfying the following conditions:\\ |
satisfying the following conditions:\\ |
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| a) $y$ depends on $n$ arbitrary constants $C_1,\,C_2,\,\ldots,\,C_n$.\\ |
a) $y$ depends on $n$ arbitrary constants $C_1,\,C_2,\,\ldots,\,C_n$.\\ |
| b) $y$ satisfies (1) with all values of $C_1,\,C_2,\,\ldots,\,C_n$\\ |
b) $y$ satisfies (1) with all values of $C_1,\,C_2,\,\ldots,\,C_n$\\ |
| c) If there are given the initial conditions\\ |
c) If there are given the initial conditions\\ |
| \qquad\qquad $y = y_0$,\,\,$y' = y_1$,\,\,$y'' = y_2$,\,\, |
\qquad\qquad $y = y_0$,\,\,$y' = y_1$,\,\,$y'' = y_2$,\,\, |
| $\ldots$,\,\,$y^{(n-1)} = y_{n-1}$\quad when\quad$x = x_0,$\\ |
$\ldots$,\,\,$y^{(n-1)} = y_{n-1}$\quad when\quad$x = x_0,$\\ |
| then one can chose the values of $C_1,\,C_2,\,\ldots,\,C_n$ such that\, |
then one can chose the values of $C_1,\,C_2,\,\ldots,\,C_n$ such that\, |
| $y = \varphi(x,\,C_1,\,C_2,\,\ldots,\,C_n)$\, fulfils those conditions (supposing that $x_0,\,y_0,\,y_1,\,y_2,\,\ldots,\,y_{n-1}$ belong to the region where the conditions for the existence of the solution are valid). |
$y = \varphi(x,\,C_1,\,C_2,\,\ldots,\,C_n)$\, fulfils those conditions (supposing that $x_0,\,y_0,\,y_1,\,y_2,\,\ldots,\,y_{n-1}$ belong to the region where the conditions for the existence of the solution are valid). |
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| Each function which is obtained from the general solution by giving certain concrete values for\, $C_1,\,C_2,\,\ldots,\,C_n$,\, is called a {\em particular solution} of (1). |
Each function which is obtained from the general solution by giving certain concrete values for\, $C_1,\,C_2,\,\ldots,\,C_n$,\, is called a {\em particular solution} of (1). |
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