for the given system of ODEs, calculate the eigenvalues and corresponding eigenvectors of the coefficient matrix (dx/dt)=2x+y (dy/dt)=x+2y


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Question Number 207096 by Wuji last updated on 06/May/24

$f o r t h e g i v e n s y s t e m o f O D E s, c a l c u l a t e t h e$ $e i g e n v a l u e s a n d c o r r e s p o n d i n g e i g e n v e c t o r s o f t h e$ $c o e f f i c i e n t m a t r i x$ $\frac{d x}{d t} = 2 x + y \frac{d y}{d t} = x + 2 y$
Commented by Wuji last updated on 07/May/24

$n e e d a h e l p i n g h a n d, p l e a s e$
Commented by aleks041103 last updated on 07/May/24

$t h e i d e a i s$ $i f q = (\begin{matrix} x \\ y \end{matrix}) a n d a l s o t h e n \frac{d q}{d t} = (\begin{matrix} d x / d t \\ d y / d t \end{matrix})$ $t h e n y o u c a n w r i t e t h e l i n e a r O D E a s$ $\frac{d q}{d t} = \hat{M} q$ $w h e r e \hat{M} i s t h e c o e f f i c i e n t m a t r i x$ $\Rightarrow i n t h i s c a s e t h e c o e f f m a t r i x i s$ $(\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}) . . .$ $t r y f r o m h e r e y o u r s e l f$
Commented by Wuji last updated on 07/May/24

$yes, sir . thank you so much$
	Answered by mr W last updated on 07/May/24

	$a l t e r n a t i v e w a y :$ $(i) + (i i) :$ $\frac{d (x + y)}{d t} = 3 (x + y)$ $\frac{d (x + y)}{x + y} = 3 d t$ $\Rightarrow \ln (x + y) = 3 t + C$ $\Rightarrow x + y = 2 C_{1} e^{3 t} . . . (I)$ $(i) - (i i) :$ $\frac{d (x - y)}{d t} = x - y$ $\frac{d (x - y)}{x - y} = d t$ $\Rightarrow \ln (x - y) = t + C$ $\Rightarrow x - y = 2 C_{2} e^{t} . . . (I I)$ $\Rightarrow x = C_{1} e^{3 t} + C_{2} e^{t}$ $\Rightarrow y = C_{1} e^{3 t} - C_{2} e^{t}$
	Commented by Wuji last updated on 07/May/24

	$God bless you, sir$
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