The first order linear differential matrix equation is
$$\frac{dY}{dt}-AY=H(t)$$
Where \(Y\) is the unknown column vector, \(H\) is the source vector, and \(A\) is a constant matrix. The solution to this equation is
$$Y(t)=exp(At)Y(0)+\int^t_0exp(At)H(t-\tau)d\tau$$
If the eigenvalues of the \(A\) matrix is distinct, the exponential is
$$exp(At)=Xexp(D)X^{-1}X(0)$$
where \(X\) is the matrix of eigenvectors of A. \(D\) is the diagonal matrix of the eigenvalues of \(A\)
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