Pure birth process

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