Calculate the generalized solution for the following system of ODEs: (dx/dt)=−(1/2)x, (dy/dt)=(1/2)x−(1/4)y, (dz/dt)=(1/4)y−(1/6)z


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Question Number 207106 by Wuji last updated on 07/May/24

$C a l c u l a t e t h e g e n e r a l i z e d s o l u t i o n f o r t h e f o l l o w i n g$ $s y s t e m o f O D E s :$ $\frac{d x}{d t} = - \frac{1}{2} x, \frac{d y}{d t} = \frac{1}{2} x - \frac{1}{4} y, \frac{d z}{d t} = \frac{1}{4} y - \frac{1}{6} z$
	Answered by mr W last updated on 07/May/24

	$\| \begin{matrix} - \frac{1}{2} - λ & 0 & 0 \\ \frac{1}{2} & - \frac{1}{4} - λ & 0 \\ 0 & \frac{1}{4} & - \frac{1}{6} - λ \end{matrix} \| = 0$ $(- \frac{1}{2} - λ) (- \frac{1}{4} - λ) (- \frac{1}{6} - λ) = 0$ $\Rightarrow λ = - \frac{1}{2}, - \frac{1}{4}, - \frac{1}{6}$ $V_{1} = [\begin{matrix} \frac{4}{3} \\ - \frac{8}{3} \\ 1 \end{matrix}]$ $V_{2} = [\begin{matrix} 0 \\ - \frac{5}{3} \\ 1 \end{matrix}]$ $V_{3} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]$ $[\begin{matrix} x \\ y \\ z \end{matrix}] = C_{1} e^{- \frac{t}{2}} [\begin{matrix} \frac{4}{3} \\ - \frac{8}{3} \\ 1 \end{matrix}] + C_{2} e^{- \frac{t}{4}} [\begin{matrix} 0 \\ - \frac{5}{3} \\ 1 \end{matrix}] + C_{3} e^{- \frac{t}{6}} [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]$
	Commented by Wuji last updated on 07/May/24

	$God bless you, sir$
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