# The coefficient matrix for a system of linear differential equations of the form y^{1} = A_{y} has the given eigenvalues

The coefficient matrix for a system of linear differential equations of the form ${y}^{1}={A}_{y}$ has the given eigenvalues and eigenspace bases. Find the general solution for the system.
$\left[{\lambda }_{1}=-1⇒\left\{\left[\begin{array}{c}103\end{array}\right]\right\},{\lambda }_{2}=3i⇒\left\{\left[\begin{array}{c}2-i1+i7i\end{array}\right]\right\},{\lambda }_{3}=-3i⇒\left\{\left[\begin{array}{c}2+i1-i-7i\end{array}\right]\right\}\right]$

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Step 1
The general solution is
$y={c}_{1}{y}_{1}+{c}_{2}{y}_{2}+{c}_{3}{y}_{3}$
for ${y}_{1}={e}^{{\lambda }_{1}t}u$
${y}_{2}={e}^{at}\left(\mathrm{sin}\left(bt\right)Re\left(u\right)+\mathrm{cos}\left(bt\right)Im\left(u\right)\right)$
${y}_{3}={e}^{at}\left(\mathrm{cos}\left(bt\right)Re\left(u\right)-\mathrm{sin}\left(bt\right)Im\left(u\right)\right)$
Step 2
We have
$\left[{\lambda }_{1}=-1⇒\left\{\left[\begin{array}{c}103\end{array}\right]\right\},{\lambda }_{2}=3i⇒\left\{\left[\begin{array}{c}2-i1+i7i\end{array}\right]\right\},{\lambda }_{3}=-3i⇒\left\{\left[\begin{array}{c}2+i1-i-7i\end{array}\right]\right\}\right]$
Step 3
Then
${y}_{1}={e}^{-t}\left(\left[\begin{array}{c}103\end{array}\right]\right)$
${y}_{2}={e}^{0t}\left(\left(\mathrm{sin}\left(3t\right)\left[\begin{array}{c}212\end{array}\right]+\mathrm{cos}\left(3t\right)\left[\begin{array}{c}-111\end{array}\right]\right)$
$=\left(\left(\mathrm{sin}\left(3t\right)\left[\begin{array}{c}212\end{array}\right]+\left(\mathrm{cos}\left(3t\right)\left[\begin{array}{c}-111\end{array}\right]\right)$
${y}_{3}={e}^{0t}\left(\mathrm{cos}\left(-3t\right)\left[\begin{array}{c}210\end{array}\right]-\mathrm{sin}\left(-3t\right)\left[\begin{array}{c}1-17\end{array}\right]\right)$
$=\left(\mathrm{cos}\left(-3t\right)\left[\begin{array}{c}210\end{array}\right]-\mathrm{sin}\left(-3t\right)\left[\begin{array}{c}1-17\end{array}\right]\right)$
Step 4
Hence the general solution is
$y={c}_{1}{y}_{1}+{c}_{2}{y}_{2}+{c}_{3}{y}_{3}$
$={c}_{1}{e}^{-t}\left(\left[\begin{array}{c}103\end{array}\right]\right)+{c}_{2}\left(\left(\mathrm{sin}\left(3t\right)\left[\begin{array}{c}212\end{array}\right]+\left(\mathrm{cos}\left(3t\right)\left[\begin{array}{c}-111\end{array}\right]\right)+{c}_{3}\left(\mathrm{cos}\left(-3t\right)\left[\begin{array}{c}210\end{array}\right]-\mathrm{sin}\left(-3t\right)\left[\begin{array}{c}1-17\end{array}\right]\right)$
Step 5
The individual functions are
${y}_{1}={c}_{1}{e}^{t}\left({c}_{2}\left(2\mathrm{sin}\left(3t\right)-\mathrm{cos}\left(3t\right)\right)+{c}_{3}\left(2\mathrm{cos}\left(-3t\right)-\mathrm{sin}\left(-3t\right)\right)\right)$