Question

Find the general solution of x1? = x1 ? 2x2, x2? = 2x1 x2 using the eigenvalue method. do not use complex exponentials in your solution.

Answer 1

Guessing on how the system is expressed:

$\begin{bmatrix}x_1\\x_2\end{bmatrix}'=\begin{bmatrix}1&-2\\2&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$

The coefficient matrix has eigenvalues $\lambda=1+2i$ with corresponding eigenvectors $\begin{bmatrix}\pm i&1\end{bmatrix}^\top$. This means the characteristic solution is

$\begin{bmatrix}x_1\\x_2\end{bmatrix}=Ce^{(1+2i)t}\begin{bmatrix}i\\1\end{bmatrix}$
$\begin{bmatrix}x_1\\x_2\end{bmatrix}=Ce^t(\cos2t+i\sin2t)\begin{bmatrix}i\\1\end{bmatrix}$
$\begin{bmatrix}x_1\\x_2\end{bmatrix}=C_1\begin{bmatrix}-e^t\sin2t\\e^t\cos2t\end{bmatrix}+C_2\begin{bmatrix}e^t\cos2t\\e^t\sin2t\end{bmatrix}$