# The coefficient matrix for a system of linear differential equations of the form y_1=Ay has the given eigenvalues and eigenspace bases. Find the gener

The coefficient matrix for a system of linear differential equations of the form ${y}_{1}=Ay$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
${\lambda }_{1}=3+i⇒\left\{\left[\begin{array}{c}2i\\ i\end{array}\right]\right\},{\lambda }_{2}=3-i⇒\left\{\left[\begin{array}{c}-2i\\ -i\end{array}\right]\right\}$
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SoosteethicU

The general solution is
$y={c}_{1}{y}_{1}+{c}_{2}{y}_{2}$

${y}_{2}={e}^{at}\left(\mathrm{sin}\left(bt\right)Re\left(u\right)+\mathrm{cos}\left(bt\right)Im\left(u\right)\right)$
We have
${\lambda }_{1}=3+i⇒\left\{\left[\begin{array}{c}2i\\ i\end{array}\right]\right\},{\lambda }_{2}=3-i⇒\left\{\left[\begin{array}{c}-2i\\ -i\end{array}\right]\right\}$
Then
${y}_{1}={e}^{3t}\left(\mathrm{cos}\left(t\right)\left[\begin{array}{c}0\\ 0\end{array}\right]-\mathrm{sin}\left(t\right)\left[\begin{array}{c}2\\ 1\end{array}\right]\right)$
${y}_{2}={e}^{t}\left(\mathrm{sin}\left(-t\right)\left[\begin{array}{c}0\\ 0\end{array}\right]+\mathrm{cos}\left(-t\right)\left[\begin{array}{c}-2\\ -1\end{array}\right]\right)$
Hence the general solution is
$y={c}_{1}{y}_{1}+{c}_{2}{y}_{2}$
$={c}_{1}{e}^{3t}\left(\mathrm{cos}\left(t\right)\left[\begin{array}{c}0\\ 0\end{array}\right]-\mathrm{sin}\left(t\right)\left[\begin{array}{c}2\\ 1\end{array}\right]\right)+{c}_{2}{e}^{t}\left(\mathrm{sin}\left(-t\right)\left[\begin{array}{c}0\\ 0\end{array}\right]+\mathrm{cos}\left(-t\right)\left[\begin{array}{c}-2\\ -1\end{array}\right]\right)$
The individual functions are
${y}_{1}=-2{c}_{1}{e}^{3t}\left(\mathrm{sin}\left(t\right)+\mathrm{cos}\left(-t\right)\right)$
${y}_{2}=-{c}_{2}{e}^{3t}\left(\mathrm{sin}\left(t\right)+\mathrm{cos}\left(-t\right)\right)$