Solve the differential system below: { ((y_1 ′=2y_1 +y_2 +y_3 )),((y_2 ′=−2y_1 −y_3 )),((y_3 ′=2y_1 +y_2 +2y


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Question Number 160451 by Ar Brandon last updated on 29/Nov/21
$Solve the differential system below: { ((y_1 ′=2y_1 +y_2 +y_3 )),((y_2 ′=−2y_1 −y_3 )),((y_3 ′=2y_1 +y_2 +2y_3 )) :}$
$Solve the differential system below :$ ${\begin{cases} y_{1}^{'} = 2 y_{1} + y_{2} + y_{3} \\ y_{2}^{'} = - 2 y_{1} - y_{3} \\ y_{3}^{'} = 2 y_{1} + y_{2} + 2 y_{3} \end{cases}$
	Answered by mr W last updated on 30/Nov/21

	$G E N E R A L M E T H O D$ $\| \begin{matrix} 2 - λ & 1 & 1 \\ - 2 & - λ & - 1 \\ 2 & 1 & 2 - λ \end{matrix} \| = 0$ $(2 - λ) (λ^{2} - 2 λ + 1) = 0$ $\Rightarrow λ_{1} = 2, λ_{2} = λ_{3} = 1$ $V_{1} = [\begin{matrix} 1 \\ - 2 \\ 2 \end{matrix}]$ $V_{2} = [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}]$ $V_{3} = [\begin{matrix} 0 \\ - 1 \\ 1 \end{matrix}]$ $\Rightarrow [\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \end{matrix}] = C_{1} e^{2 x} [\begin{matrix} 1 \\ - 2 \\ 2 \end{matrix}] + C_{2} e^{x} [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}] + C_{3} e^{x} [\begin{matrix} 0 \\ - 1 \\ 1 \end{matrix}]$ $o r$ $y_{1} = C_{1} e^{2 x}$ $y_{2} = - 2 C_{1} e^{2 x} + C_{2} e^{x}$ $y_{3} = - y_{2}$
	Commented by Ar Brandon last updated on 30/Nov/21

	$Thank you Sir$
	Commented by mr W last updated on 30/Nov/21

	$i n e e d t o c h e c k t h e w o r k i n g, m a y b e$ $t h e r e i s a m i s t a k e i n s i d e .$
	Answered by mr W last updated on 30/Nov/21

	$a n o t h e r m e t h o d$ $(i i) + (i i i) :$ $y_{2}^{'} + y_{3}^{^{'}} = y_{2} + y_{3}$ ${(y_{2} + y_{3})}^{'} = y_{2} + y_{3}$ $\Rightarrow y_{2} + y_{3} = C_{1} e^{x}$ $y_{1}^{'} = 2 y_{1} + C_{1} e^{x}$ $\Rightarrow y_{1} = C_{2} e^{2 x} - C_{1} e^{x}$ $y_{2}^{'} - y_{2} = - 2 y_{1} - (y_{3} + y_{2})$ $y_{2}^{'} - y_{2} = - 2 C_{2} e^{2 x} + C_{1} e^{x}$ $\Rightarrow y_{2} = (C_{1} x + C_{3}) e^{x} - 2 C_{2} e^{2 x}$ $\Rightarrow y_{3} = (C_{1} - C_{1} x - C_{3}) e^{x} + 2 C_{2} e^{2 x}$
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