ODELS2

Find a fundamental matrix and the Wronskian for the following linear system: $\begin{cases}x_1'=x_2 \\ x_2'=x_1\end{cases}\,$ .

The matrix of coefficients $A\,$ is $\begin{bmatrix} 0&1\\1&0 \end{bmatrix}\,$ .

The eigenvalues are $\lambda=\pm 1\,$ .

For $\lambda=1\,$ an eigenvector is $u_1=\begin{bmatrix}1\\1\end{bmatrix}\,$ .

For $\lambda=-1\,$ an eignvector is $u_{-1}=\begin{bmatrix}-1\\1\end{bmatrix}\,$ .

Therefore two linearly independent solutions are:

$x^1=e^t\begin{bmatrix}1\\1\end{bmatrix} = \begin{bmatrix} e^t \\ e^t \end{bmatrix}\,$

$x^2=e^{-t}\begin{bmatrix}-1\\1\end{bmatrix} = \begin{bmatrix} -e^{-t} \\ e^{-t} \end{bmatrix}\,$

So a fundamental matrix is $X(t)=\begin{bmatrix} e^t & -e^{-t} \\ e^t & e^{-t} \end{bmatrix}\,$ .

The Wronskian is the determinant of this matrix after evaluation at $t=0\,$ .

$W=1+1=2\,$