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Laplace-transform solution of a constant-coefficient vector ODE

For

y^{'} = A y + g, qq u a d y (0) = y_{0},

the derivative rule gives

(s I - A) Y (s) = y_{0} + G (s),

and hence

Y (s) = (s I - A)^{- 1} (y_{0} + G (s))

wherever the transforms converge and

s I - A

is invertible.

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