The N point function is simply

$$\frac{⟨\mathrm{\Omega }|T[\varphi (x_1)\varphi (x_2)\varphi (x_3)...\varphi (x_n)]|\mathrm{\Omega }⟩}{⟨\mathrm{\Omega }||\mathrm{\Omega }⟩}$$

which can be written in free fields and free vacuum by the following relations.

$$\begin{array}{r}{\varphi }_0(\overrightarrow{x},t)=e^{i{H}_{0}t}\varphi (\overrightarrow{x},0)e^{-i{H}_{0}t}\\ {\varphi }_{(}\overrightarrow{x},t)=e^{iHt}\varphi (\overrightarrow{x},0)e^{-iHt}\\ \text{above equation implies that}\\ {\varphi }_{(}\overrightarrow{x},t)& =e^{iHt}{e}^{-i{H}_{0}t}{\varphi }_0(\overrightarrow{x},t)e^{i{H}_{0}t}{e}^{-iHt}\\ & =U^{†}(t,0){\varphi }_0(\overrightarrow{x},t)U(t,0)\\ \text{where ,}\\ U(t,0)=e^{i{H}_{0}t}{e}^{-iHt}\end{array}$$

We start looking for some differential equation for U(t,0), and we get :

$$\begin{array}{rl}\frac{dU}{dt}& =i{H}_{0}U-iUH\\ & =i{e}^{i{H}_{0}t}(H_0-H)e^{-iHt}\\ & =-i{e}^{i{H}_{0}t}(H_{int})e^{-iHt}\\ & =-i(e^{i{H}_{0}t}(H_{int})e^{-i{H}_{0}t})e^{i{H}_{0}t}{e}^{-iHt}\\ & =-i{H}_I(t)U(t,0)\\ \\ \boxed{{\displaystyle \frac{dU}{dt}=-i{H}_I(t)U(t,0)}}\\ \end{array}$$