10.3 Analytical Solution for Linear Coupled Differential Equation

Use Laplace transformation and Cramer’s rule

Using matrix exponential

\[\begin{align*} \frac{dx}{dt} &= -k x \\ x &= x_0 e^{-kt} \\ \frac{d\vec{x}}{dt} &= A \vec{x} \\ \vec{x} &= \vec{x}_0 e^{At} \end{align*}\]

See “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later.PDF”