In Ralph C. Smith’s book, he talks about three ways to calculate the sensitivity matrix for a problem. You can solve the problem analytically and differentiate the solution with respect to the parameters, you can calculate the solution numerically after a small step on the parameter space then use finite difference; or you can differentiate the initial dynamic with respect to the parameters and solve the new system analytically again (that’s what he calls sensitivity equations). He provides an example of the third one in chapter 14, but the problem does not have analytic solution, image below.

In chapter 7, he talks about a similar toy model, and asks you to solve the sensitivity equation analytically. My question is that, by solving the sensitivity equation, I would’ve to use the equation for the initial dynamic in my ODE system, then I would eventually solve the problem as a whole, which makes using this kind of approach senseless. For the toy model I’m referring, see image below.

The only way that I can see to get around this is if there was a way to solve the sensitivity equation without using the initial dynamic in my ODE system, but I can’t see a way to do this.

Any help would be appreciated, thanks!