 # 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 Tolnaio 2021-01-23 Answered
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
${\lambda }_{1}=2i⇒\left\{\left[\begin{array}{c}1+i\\ 2-i\end{array}\right]\right\},{\lambda }_{2}=-2i⇒\left\{\left[\begin{array}{c}1-i\\ 2+i\end{array}\right]\right\}$
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By theorem 6.19 we know that the solution is
$y={c}_{1}{e}^{{\lambda }_{1}t}{u}_{1}+\dots +{c}_{n}{e}^{{\lambda }_{n}t}{u}_{n}$
with ${\lambda }_{i}$ the eigenvalues of the matrix A nad ${u}_{i}$ the eigenvectors Thus for this case we then obtain the general solution:
$\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\end{array}\right]=y={c}_{1}{e}^{2it}\left[\begin{array}{c}1+i\\ 2-i\end{array}\right]+{c}_{2}{e}^{-2it}\left[\begin{array}{c}1-i\\ 2+i\end{array}\right]$
Thus we obtain:
${y}_{1}={c}_{1}\left(\mathrm{cos}\left(2t\right)-\mathrm{sin}\left(2t\right)\right)+{c}_{2}\left(\mathrm{cos}\left(2t\right)+\mathrm{sin}\left(2t\right)\right)$
${y}_{2}={c}_{1}\left(2\mathrm{cos}\left(2t\right)+\mathrm{sin}\left(2t\right)\right)+{c}_{2}\left(\mathrm{cos}\left(2t\right)-2\mathrm{sin}\left(2t\right)\right)$