I have the following system of four linear coupled ODEs, where the dot indicates derivative with respect to time. The T's are the main variables, P is an input (known) and R and C are constants.
\begin{equation} C_t.\dot{T}_t = \frac{T_c - T_t}{R_4} + \frac{T_h - T_t}{R_5} + P_t \end{equation}
\begin{equation} C_h.\dot{T}_h = \frac{T_b - T_h}{R_2} + \frac{T_c - T_h}{R_1} + \frac{T_t - T_h}{R_5} \end{equation}
\begin{equation}\label{eq:2.13} C_c.\dot{T}_c = \frac{T_b - T_c}{R_3} + \frac{T_t - T_c}{R_4} + \frac{T_h - T_c}{R_1} + P_c \end{equation}
\begin{equation}\label{eq:2.14} C_b.\dot{T}_b = \frac{T_k - T_b}{R_6} + \frac{T_c - T_b}{R_3} + \frac{T_h - T_b}{R_2} \end{equation}
I would like to calculate
\begin{equation} \frac{\partial T_t}{\partial R_6} \end{equation}
In other words, how would $T_t$ change is the constant $R_6$ is changed.
but I am not sure how to do it, I can try discretizing the equations and then finding a solution for each expression. Any idea how I can do it?
\begin{equation} C_t.(Tt_ {t+1}-Tt_ {t} )/\Delta t= \frac{Tc_t - Tt_t}{R_4} + \frac{Th_t - Tt_t}{R_5} + P_t \end{equation}
Subscript t+1 is for the next time step and subscript t is for the current value. That way I can solve for each T variable and somehow use the chain rule to get the value using.
\begin{equation} \frac{\partial T_t}{\partial R_6} = \frac{\partial y}{\partial T_t} \frac{d T_t}{\partial T_c} \frac{d T_c}{\partial T_h} \frac{d T_h}{\partial T_b} \frac{d T_b}{\partial R_6} \end{equation}
Where the rate of change of $T_{t+1}$ is calculated based on knowledge of $T_{t}$. I am not sure if this is valid, please help.
Thanks