Hallo,
ich habe erhalten:
\( \operatorname{det}(A-\boldsymbol{\lambda} I)=\left|\begin{array}{cc}-\boldsymbol{\lambda}-\mathbf{3} & -\mathbf{1} \\ \mathbf{2} & -\boldsymbol{\lambda}-\mathbf{1}\end{array}\right|=\mathbf{0} \)
\( \lambda^{2}+\mathbf{4} \boldsymbol{\lambda}+\mathbf{5} \rightarrow \lambda_{1,2}= \pm i-2 \)
Eigenwert Eigenvektoren:
-2-i \( \begin{pmatrix} -1-i\\1\\ \end{pmatrix} \)
-2+i \( \begin{pmatrix} -1+i\\1\\ \end{pmatrix} \)
\( y=C_{1} e^{-2 t}\left(\cos (t)\left(\begin{array}{c}-1 \\ 2\end{array}\right)-\sin (t)\left(\begin{array}{l}1 \\ 0\end{array}\right)\right)+C_{2} e^{-2 t}\left(\cos (t)\left(\begin{array}{l}1 \\ 0\end{array}\right)+\sin (t)\left(\begin{array}{c}-1 \\ 2\end{array}\right)\right) \)