I'm studing the Poisson Process and i have two definitions. I got stuck on a step to proove that the two definitions are equivalent.
Can i claim this:
The solution to this differential equation is: $P_{0}(t) = e^{-\alpha t}$
$\dfrac{\dot{P}_{0}(t)}{P_{0}(t)} = \alpha$ with the initial condition: $P_{0}(0) = 1$ and $\alpha$ is a constant.
Is it true? My intuition is Yes, and correct me if i'm wrong, but the only function that satisfies that his derivative divided by the function equals a constant is the exponential function.
Thank's..
Your solution is wrong. True is $$ \frac{d}{dt}(e^{-αt}P(t))=e^{-αt}(\dot P(t)-αP(t)) = 0 $$ so that $$ e^{-αt}P(t)=const.=P(0) $$ so that $\dot P=αP$ has the only solution $P(t)=e^{αt}P(0)$.
You would need to change the sign before $α$ to get the opposite case.