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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  contains only constants and  is function of  .
One way to solve these n coupled first order linear ODE is to diagonalize the coefficient matrix and thus decouple these equations.
Suppose that  and  are distinct eigenvalues and associated eigenvectors of
Let  and
the system of differential equations can be rewritten as
Move  to the right hand side of the equation, we have
where
and
The general solution of this system of equations
is
Substituting back to  , the original system of differential equations can be solved:
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