Differential equation for $F_n$

We have

$$I_n(x) = F_n(x) {\rm e}^x - (-1)^n n!$$

Differentiating both sides of the equation gives

$$x^n {\rm e}^x = F_n'(x) {\rm e}^x + F_n(x) {\rm e}^x$$

Hence, $F_n(x)$ satisfies

$$y' + y = x^n$$ (3)

We will use this to deduce an expression for $F_n(x)$.