The general solution of the system $\dot{X} = (0 -1 \ 1 0)X + (\cos t - \sin t)$ at $t_0 = 0$ is $X_p(t) = \begin{pmatrix} e^t \\ e^{-t} \end{pmatrix} \int_0^t e^{-s} (\cos s - \sin s) \, ds$.
The first step is to find the fundamental matrix $\Phi(t)$ of the system $\dot{X} = (0 -1 \ 1 0)X$. This can be done by finding the eigenvalues and eigenvectors of the matrix $A = (0 -1 \ 1 0)$.
The eigenvalues of $A$ are $0$ and $1$, and the corresponding eigenvectors are $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Therefore, the fundamental matrix is \Phi(t) = \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
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Now, we can use the result in the problem statement to find the solution of the inhomogeneous system $\dot{X} = (0 -1 \ 1 0)X + (\cos t - \sin t)$. The solution is X_p(t) = \Phi(t) \int_0^t \Phi^{-1}(s) (\cos s - \sin s) \, ds
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Plugging in the expression for $\Phi(t)$, we get
X_p(t) = \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix} \int_0^t \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} (\cos s - \sin s) \, ds
Evaluating the integral, we get
X_p(t) = \begin{pmatrix} e^t \\ e^{-t} \end{pmatrix} \int_0^t e^{-s} (\cos s - \sin s) \, ds
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This is the general solution of the system $\dot{X} = (0 -1 \ 1 0)X + (\cos t - \sin t)$ at $t_0 = 0$.
The fundamental matrix $\Phi(t)$ is the matrix that satisfies the following differential equation: \dot{\Phi}(t) = A \Phi(t)
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with the initial condition $\Phi(0) = I$. The solution of the inhomogeneous system $\dot{X} = AX + F$ can be written as X_p(t) = \Phi(t) \int_0^t \Phi^{-1}(s) F(s) \, ds, where $F(t)$ is the inhomogeneous term.