for-the-given-system-of-ODEs-calculate-the-eigenvalues-and-corresponding-eigenvectors-of-the-coefficient-matrix-dx-dt-2x-y-dy-dt-x-2y-

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}) \dots

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} \dots (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} \dots (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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