I am having hard time to convert following set of differential equation to state space equation. I am a biologist and my math skills fall short as I don't know where to start. Any suggestion or feedback is highly appreciated. Thanks in advance.

**Updates**:
Based on comments I have updated the question (which is not a good practice, my sincere apologies to everyone),

I am modelling biological system with a cascade of signal transduction steps with feedback loops. Each level in the cascade has corresponding equation $\dot{y_i}$ or $dy_i/dt$ given by
\begin{equation*}
\frac{dy_i}{dt}= g_i(v_i, y_i)+ c_i + e_i
\end{equation*}
For example
\begin{equation*}
\frac{dy_0}{dt}= g_0(a_{00}y_0+a_{02}y_2, y_0)+ c_0
\end{equation*}
\begin{equation*}
\frac{dy_1}{dt}= g_1(a_{10}y_0+a_{11}y_1+a_{12}y_2, y_1)\\
\end{equation*}
\begin{equation*}
\frac{dy_2}{dt}= g_2(a_{20}y_0+a_{21}y_1+a_{22}y_2+a_{23}y_3+a_{24}y_4+a_{25}y_5, y_2)+e_2
\end{equation*}
\begin{equation*}
\frac{dy_3}{dt}= g_3(a_{32}y_2+a_{33}y_3, y_3)
\end{equation*}
\begin{equation*}
\frac{dy_4}{dt}= g_4(a_{42}y_2+a_{44}y_4, y_4)
\end{equation*}
\begin{equation*}
\frac{dy_5}{dt}= g_5(a_{25}x_2-a_{25}x_5)
\end{equation*}

and so on. $y_i$ represents state variable and only $y_5$ can be observed ("output") others are hidden. $e_2$ is input variable. $a_{jk}$ represents parameters. Function $g_i$ is given by multiplication of $h_i$ and $r_i$
\begin{equation*}
g_i({v_i}, y_i)= h_i ({v_i})\cdot r_i({v_i}, y_i)\\
\end{equation*}
where function $h_i$ and $r_i$ are given by

\begin{equation*}
h_i({v_i}) = \begin{cases} \frac{{v_i}}{1+\frac{{v_i}}{S_i}(1-exp(-{v_i}/S_i))} & \mbox{when } {v_i}> 0, \cr {v_i} & \mbox{when } {v_i}\leq 0, \end{cases}
\end{equation*}
\begin{equation*}
r_i({v_i}, y) = \begin{cases} 1-exp(\frac{{v_i}^2S_i}{{v_i}(\varepsilon -y)^2}) & \mbox{when } y<\varepsilon\ \&\ {v_i}< 0, \cr 1 & \mbox{otherwise.} \end{cases}
\end{equation*}

where

Function $h_i$ puts physiologically relevant soft upper limit.

Function $r_i$ ensures non-negative ligand concentration.

$\dot v$ is corresponding linear rate.

$S_i$ is max size of the pool.

$\varepsilon$ is a small positive constant.

**Update-2:**
Ok, I want to know if I am doing right thing here.
I need state space equation for my system of differential equation with nonlinear function. Technically I can have two different versions

**A non-linear version**
\begin{equation*}
\frac{dy_i}{dt}= g_i(v_i, y_i)+ c_i + e_i
\end{equation*}

**A linear version**
\begin{equation*}
\frac{dy_i}{dt}= v_i+ c_i + e_i
\end{equation*}

For linear version I can write the state equation as $y'= Ay+c+e$ (see the matrix equation as Image)

Taking this to next step can I write like following?

$y'= g(Ay)+c+e$

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