Number of arrivals distribution, Poisson process

Question

Suppose that arrivals occur at $T_1, T_2, \ldots$ . The interval times $A_n = T_n - T_{n-1}$ are i.i.d. $\exp(\lambda)$, where $T_0 =0$. Let $N(t)$ denote the number of arrivals in $[0,t]$ and define for $n \geq 0$ \begin{align} p_n(t) = \mathbb{P}(N(t)=n), \qquad t>0. \end{align} It could be shown that for $n \geq 1$ \begin{align} p_n(t) = \int_0^t p_{n-1}(t-x) \lambda e^{-\lambda x}\ dx, \qquad t>0. \qquad (*) \end{align} In order to find the distribution of $p_n$, I want to solve this integral equation. I am aware of the fact that $p_n$ satisfies the following differential equation \begin{align} p_n'(t) = -\lambda p_n(t) + \lambda p_{n-1}(t), \qquad t>0, \end{align} with initial condition $p_n(0)=0$. Even considering implementing the above expression in $(*)$, I do not see how to solve the integral. Any help is appreciated!