I have the following set of differential-difference equations $$ \begin{aligned} & (s F(s))^{\prime}=f(s-1) \text { for } s>1 \\ & (s f(s))^{\prime}=F(s-1) \text { for } s>2 \end{aligned}$$ with initial conditions $F(s)=\frac{2 e^\gamma}{s}$ for $1 \leq s \leq 3, f(s)=0$ for $1 \leq s \leq 2$
I want to show that the equations above are satisfied by $$ \begin{gathered} F(s):=2 e^\gamma\left(\frac{\mathbb{1}_{s>1}}{s}+\sum_{j \geq 3, \text { odd }} \frac{1}{j !} \int_{[1,+\infty)^{j-1}} \mathbb{1}_{t_1+\ldots t_{j-1} \leq s-1} \frac{d t_1 \ldots d t_{j-1}}{t_1 \ldots t_j}\right) \\ f(s):=2 e^\gamma \sum_{j \geq 2, \text { even }} \frac{1}{j !} \int_{[1,+\infty)^{j-1}} \mathbb{1}_{t_1+\ldots +t_{j-1} \leq s-1} \frac{d t_1 \ldots d t_{j-1}}{t_1 \ldots t_j} \end{gathered} $$ 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.