I am involved in sensitivity analysis of an ODE system. The sensitivity $Z_i$ is defined as $$Z_i:=\frac{\partial y_i}{\partial c}$$ for a state function $y_i$ and a parameter $c$.
(In this post, I want to iterate through multiple state functions but stay with one parameter. It should be possible to repeat the process for different $c$.)
The temporary derivative of $y_i$ shall be $\dot{y}_i$ which is described via the function $$\dot{y}_i=f_i(y,P,t)$$ with the set of all parameters $P$ and time $t$ as well as the vector of all state functions $y$.
I am looking for the solution of the following ODE system: $$\dot{Z}=f_c+JZ$$ $f_c$ is a vector with components $$f_{c,i}=\frac{\partial f_i}{\partial c}$$ with one parameter $c$. $J$ shall be the Jacobian with the derivatives of each $f_i$ considering $y_j$ (former $y_i$).
Is it true that the solution is $$ \left ( \begin{matrix} Z_1 \\ ... \\ Z_I \end{matrix} \right ) = \mathbf{J}^{-1}\cdot \exp\left ( (\mathbf{J}\cdot t) - \mathbb{I} \right )\cdot \left ( \begin{matrix} \frac{\partial f_1}{\partial c} \\ ... \\ \frac{\partial f_I}{\partial c} \end{matrix} \right ) $$ with identity/unit matrix $\mathbb{I}$ and former defined $i\in I$?