Question: Recalling that P0(t) = e-t, check that P1(t) = te-t is a solution of your equation with initial condition P1(0) = 0. The probabilities for

Recalling that P0(t) = e-λt, check that P1(t) = λte-λt is a solution of your equation with initial condition P1(0) = 0.
The probabilities for the Poisson distribution can be derived by solving differential equations. Let Pi (t) be the probability of exactly i events by time t, assuming an underlying rate of λ.

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