Let X(t) be a pure birth process with X(0) = 1 and parameters \lambda_{k}=k Let p_{n}(t)=P(X(t)=n) show that p_{n}(t) satisfy the equation p'_{n}=(n-1)p_{n-1}(t)-np_{n}(t) for all n\geq1

How do i work out what is p_{n}(t)=P(X(t)=n) in terms of exponentials. Can you help please? thanks