# Besides stiffness problem, the explicit methods are still not better than implicit methods?

Considering the simple dynamical system as follows

${\bf C}{\bf \dot{x}}(t)={\bf G}{\bf }x(t)+{\bf B}{\bf u}(t)$

where $\bf{C}$ and $\bf{G}$ are sparse matrices, ${\bf u}(t)$ is a input and $\bf{B}$ is incident matrix which poses ${\bf u}(t)$ in the system.

In my thought, the explicit methods work if there is not $\bf{C}$ on the left hand side,

${\bf \dot{x}}(t)={\bf G}{\bf x}(t)+{\bf B}{\bf u}(t)$

e.g. solve it by forward Euler method and ignore the input for simplicity ($\bf{\dot{x}}(t)={\bf G}{\bf x}(t))$), then

${\bf x}(t+h)={\bf x}(t)+h{\bf G}{\bf x}(t)$, where $h$ is the discretized time step.

However, when this is not true, the explicit method still need to solve linear system, because

${\bf C}{\bf x}(t+h)={\bf C}{\bf x}(t)+h{\bf G}{\bf x}(t)$ or say it requires inversion of matrix ${\bf x}(t+h)={\bf x}(t)+h{\bf C}^{-1}{\bf G}{\bf x}(t)$