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A system of first order linear ordinary differential equation can be expressed as the following form  or in the matrix form  where the matrix One way to solve these n coupled first order linear ODE is to diagonalize the coefficient matrix Suppose that  Let  the system of differential equations can be rewritten as  Move  where     , the original system of differential equations can be solved:
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contains only constants and 
is function of 
.
and 
are distinct eigenvalues and associated eigenvectors of
and
to the right hand side of the equation, we have
