Let $$ \mathrm{y}^{\prime}= $$ Ay be a system of differential equations where $$ A=\left[\begin{array}{cc}-5 & 1 \ 0 & -5\end{array} ight] $$, The matrix has spectrum $$ \lambda(A)=\{-5\} $$ and $$ \mathbf{c}=\left[\begin{array}{l}5 \ 0\end{array} ight] $$ is an eigenvector of $$ A $$ corresponding to $$ r=-5 $$, The vector $$ \mathrm{d}=\left[\begin{array}{c}-3 \ 5\end{array} ight] $$ satisfies the equation $$ (A-r I) \mathrm{d}=\mathbf{c} $$. What is the general solution to the system of differential equations? $$ \left[\begin{array}{l}y_{1} \ y_{2}\end{array} ight]=k_{1} e $$

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Let $$ \mathrm{y}^{\prime}= $$ Ay be a system of differential equations where $$ A=\left[\begin{array}{cc}-5 & 1 \ 0 & -5\end{array}ight] $$, The matrix has spectrum $$ \lambda(A)=\{-5\} $$ and $$ \mathbf{c}=\left[\begin{array}{l}5 \ 0\end{array}ight] $$ is an eigenvector of $$ A $$ corresponding to $$ r=-5 $$, The vector $$ \mathrm{d}=\left[\begin{array}{c}-3 \ 5\end{array}ight] $$ satisfies the equation $$ (A-r I) \mathrm{d}=\mathbf{c} $$. What is the general solution to the system of differential equations? $$ \left[\begin{array}{l}y_{1} \ y_{2}\end{array}ight]=k_{1} e $$

Crawford Sherman · Accepted Answer

The general solution to the system is: $$ \mathbf{y}(t) = k_1 e^{-5t} \begin{bmatrix} 5 \ 0 \end{bmatrix} + k_2 e^{-5t} \begin{bmatrix} -3 + 5t \ 5 \end{bmatrix} $$ where $$ k_1 $$ and $$ k_2 $$ are arbitrary constants.

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