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

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