$$\frac{\mathrm dw(t)}{\mathrm dt}+2w(t)=y(t)$$ $$\frac{\mathrm dy(t)}{\mathrm dt}+3y(t)=2w(t)+f(t)$$ The input to the system is $~f(t)$ and the output is $~y(t).$
The initial conditions are $~w(0)=0,\quad y(0)=1$.
Write a state space formulation of this system and solve using the
state space methods for an input $~f(t)=\delta(t)$
solving the system gives
w = -e^-t + e^t
y = -e^-t + 3e^t
This solution checks out when plugged into the first equation, but I am unsure of what to do with the second equation to check my work. Anywho, why not plug it in? I get
10e^t = f
Either I did the problem right and don't know how to check it or I did the problem wrong. In order to match the f, I need a delta function in there somewhere.