I have the following set of differential-difference equations related to Buchstab iterations in number theory $$ \begin{aligned} & (s F(s))^{\prime}=f(s-1) \text { for } s>1 \\ & (s f(s))^{\prime}=F(s-1) \text { for } s>2 \end{aligned} $$ with initial conditions $F(s)=\frac{2 e^\gamma}{s}$ for $1 \leq s \leq 3, f(s)=0$ for $1 \leq s \leq 2$, where $\gamma$ is a constant.
I want to show that the equations above are satisfied by $$ F(s):=2 e^\gamma\left(\frac{1_{s>1}}{s}+\sum_{j \geq 3, \text { odd }} \frac{1}{j !} \int_{[1,+\infty)^{j-1}} 1_{t_1+\ldots t_{j-1} \leq s-1} \frac{d t_1 \ldots d t_{j-1}}{t_1 \ldots t_j}\right) $$ $$ f(s):=2 e^\gamma \sum_{j \geq 2, \text { even }} \frac{1}{j !} \int_{[1,+\infty)^{j-1}} 1_{t_1+\ldots +t_{j-1} \leq s-1} \frac{d t_1 \ldots d t_{j-1}}{t_1 \ldots t_j} $$ with the convention that $t_j:=s-t_1-\cdots-t_{j-1}$.
I am struggling to see how I can differentiate when the indicator is under the integral. I have tried a Dirac delta approach, as well as a recursive argument but I didn't get anywhere. Any pointers would appreciated.