ODELS3

Find a fundamental matrix and the Wronskian for the following linear system: $\begin{cases}x_1'=(\sin t) x_2 \\ x_2'=-x_2\end{cases}\,$ .

Solve the easy ODE for $x_2: x_2=c_1e^{-t}\,$ .

Now

$x_1'=(\sin t)c_1e^{-t}\,$

$x_1 = c_1 \int_0^t \sin\tau e^{-\tau}d\tau = \frac{-c_1}{2} e^{-t}(\cos t + \sin t) + c_2\,$

So the fundamental matrix is the matrix of coefficients of $c_1\,$ and $c_2\,$ .

$X = \begin{bmatrix} \frac{-1}{2}e^{-t}(\cos t + \sin t) & 1 \\ e^{-t} & 0 \end{bmatrix}\,$

The Wronskian $W\,$ is the determinant of $X\,$ :

$W=\det(X)=-e^{-t}\,$