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Question Number 180718 by a.lgnaoui last updated on 16/Nov/22

$$Resoudrelesysteme$$$$\frac{dx}{dt}=4x+6y$$$$\frac{dy}{dt}=-3x-5y$$$$\frac{dz}{dt}=-3x-6y-5z$$

Answered by mr W last updated on 16/Nov/22

$$\left|\begin{array}{cc}4-\lambda & 6\\ -3& -5-\lambda \end{array}\right|=0$$$$-(4-\lambda )(5+\lambda )+3\times 6=0$$$$(\lambda +2)(\lambda -1)=0$$$$\Rightarrow \lambda =-2,1$$$$V_1=\left[\begin{array}{c}1\\ -1\end{array}\right]$$$$V_2=\left[\begin{array}{c}2\\ -1\end{array}\right]$$$$\Rightarrow x=c_1{e}^{-2t}+2c_2{e}^t$$$$\Rightarrow y=-c_1{e}^{-2t}-c_2{e}^t$$$$\frac{dz}{dt}+5z=3c_1{e}^{-2t}$$$$\Rightarrow z=\frac{\int 3c_1{e}^{-2t}{e}^{5t}dt}{e^{5t}}+c_3$$$$\Rightarrow z=c_1{e}^{-2t}+c_3$$

Commented by a.lgnaoui last updated on 24/Nov/22

$$thankyouverymuch$$

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