# How to solve the ODE system for sensitivity analysis in a dynamical system?

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 $$i\in I$$?

• Only if $J$ and $f_c$ are constant. If not the solution formula can be almost anything. May 31, 2019 at 8:19
• Okay, I see. You are right! But it is surely true for constants $J,f_c$, right? May 31, 2019 at 8:47
• Yes. One relatively easy way to see this is to insert a power series for $Z$ and then compare coefficients to get an iterative formula. In the power series, the $J^{-1}$ factor cancels, as there is no constant term in $e^{Jt}-I$. May 31, 2019 at 9:07

Thanks to @LutzL, I recognized that the given solution is a solution of the ODE problem above but only for $$f_c,J=\text{const.}$$. Thus, they aren't allowed to be functions of time.