Find the exact solution of a simple ODE system with unknown parameters Here is an ODE system:
$
  x' =
   \left(\begin{array}{*5{c}}
    -\lambda_1 & 0 & 0 & 0 & 0\\
    \lambda_1 & -(\lambda_2+\lambda_3) & 0 & 0 & 0\\
    0 & \lambda_2 & 0 & 0 & 0\\
    0 & \lambda_3 & 0 & -\lambda_2 & 0\\
    0 & 0 & 0 & \lambda_2 & 0\\
  \end{array}\right) x
$ $\hspace{4cm}$     with $x(0)=\left(\begin{array}{*1{c}}
   1\\
   0\\
   0\\
   0\\
   0\\
  \end{array}\right)$
We know that the three parameters $\lambda_1$, $\lambda_2$, $\lambda_3$ are strictly positive.
Is there any method to find the exact solution $x(t)$?
I know that this solution exists whatever the value of these three parameters, but I don't know how to solve it when the numeric values of these parameters are not known.
 A: This matrix is lower triangular, so its eigenvalues are the diagonal elements: $0$ (with multiplicity $2$), $-\lambda_1$, $-\lambda_2$ and $-\lambda_2-\lambda_3$.  The eigenvectors are easy to compute.
However, watch out for the cases $\lambda_1 = \lambda_2$ and $\lambda_1 = \lambda_2 + \lambda_3$.  With the exception of those cases, you get
$$ x =  \left[ \begin {array}{c} {{\rm e}^{-\lambda_{{1}}t}}
\\ -{\frac {\lambda_{{1}} \left( {{\rm e}^{-\lambda_
{{1}}t}}-{{\rm e}^{- \left( \lambda_{{2}}+\lambda_{{3}} \right) t}}
 \right) }{-\lambda_{{2}}-\lambda_{{3}}+\lambda_{{1}}}}
\\ {\frac { \left( {{\rm e}^{-\lambda_{{1}}t}}
\lambda_{{2}}+\lambda_{{3}}{{\rm e}^{-\lambda_{{1}}t}}-{{\rm e}^{-
 \left( \lambda_{{2}}+\lambda_{{3}} \right) t}}\lambda_{{1}}+\lambda_{
{1}}-\lambda_{{2}}-\lambda_{{3}} \right) \lambda_{{2}}}{ \left( 
\lambda_{{2}}+\lambda_{{3}} \right)  \left( -\lambda_{{2}}-\lambda_{{3
}}+\lambda_{{1}} \right) }}\\ {\frac { \left( {
{\rm e}^{-\lambda_{{2}}t}}\lambda_{{1}}-{{\rm e}^{-\lambda_{{2}}t}}
\lambda_{{2}}-{{\rm e}^{-\lambda_{{2}}t}}\lambda_{{3}}+\lambda_{{3}}{
{\rm e}^{-\lambda_{{1}}t}}-{{\rm e}^{- \left( \lambda_{{2}}+\lambda_{{
3}} \right) t}}\lambda_{{1}}+{{\rm e}^{- \left( \lambda_{{2}}+\lambda_
{{3}} \right) t}}\lambda_{{2}} \right) \lambda_{{1}}}{ \left( \lambda_
{{1}}-\lambda_{{2}} \right)  \left( -\lambda_{{2}}-\lambda_{{3}}+
\lambda_{{1}} \right) }}\\ -{\frac {{{\rm e}^{-
\lambda_{{2}}t}}{\lambda_{{1}}}^{2}\lambda_{{2}}+{{\rm e}^{-\lambda_{{
2}}t}}{\lambda_{{1}}}^{2}\lambda_{{3}}-{{\rm e}^{-\lambda_{{2}}t}}
\lambda_{{1}}{\lambda_{{2}}}^{2}-2\,{{\rm e}^{-\lambda_{{2}}t}}\lambda
_{{1}}\lambda_{{2}}\lambda_{{3}}-{{\rm e}^{-\lambda_{{2}}t}}\lambda_{{
1}}{\lambda_{{3}}}^{2}+{{\rm e}^{-\lambda_{{1}}t}}{\lambda_{{2}}}^{2}
\lambda_{{3}}+{{\rm e}^{-\lambda_{{1}}t}}\lambda_{{2}}{\lambda_{{3}}}^
{2}-{\lambda_{{1}}}^{2}\lambda_{{2}}{{\rm e}^{- \left( \lambda_{{2}}+
\lambda_{{3}} \right) t}}+\lambda_{{1}}{\lambda_{{2}}}^{2}{{\rm e}^{-
 \left( \lambda_{{2}}+\lambda_{{3}} \right) t}}-{\lambda_{{1}}}^{2}
\lambda_{{3}}+2\,\lambda_{{1}}\lambda_{{2}}\lambda_{{3}}+\lambda_{{1}}
{\lambda_{{3}}}^{2}-{\lambda_{{2}}}^{2}\lambda_{{3}}-\lambda_{{2}}{
\lambda_{{3}}}^{2}}{ \left( \lambda_{{2}}+\lambda_{{3}} \right) 
 \left( -\lambda_{{2}}-\lambda_{{3}}+\lambda_{{1}} \right)  \left( 
\lambda_{{1}}-\lambda_{{2}} \right) }}\end {array} \right] 
$$
