Kalman Filter with single input?
The Kalman filter is the least-squares optimal solution for the states of a set of linear dynamic equations under a Gaussian additive noise assumption. Hence, if you have an unforced linear first-order differential equation you can you use the Kalman filter to estimate the state of this differential equation. I'm not sure it will be very useful though, since the convergence rate of the observer has to be much faster than the dynamics of the first-order system in order to capture most of the transient response, which is the only response in an unforced system,  and to converge if the system is unstable (unless you know the initial state, which is rare). High gain amplifies noise, so you might end up with a poor estimate. For such a simple system you might want to use bandwidth-limited numerical differentiation instead to find the inverse of the filter/system to find the state.