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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_{1}e^{{-t}}\,.


x_{1}'=(\sin t)c_{1}e^{{-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\,:


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