Thanks to sound remarks here and here, and looking again at these equations, I noticed that my puzzle boils down to these 2 special functions: $$\int_0^x I_i(u) e^{-a u}du, \quad i=0,1$$ where $I_n(u)$ is the modified Bessel of the first kind.

Are these integral echoing something known in the literature? What can we know for $x<+\infty$ and $\alpha >1$? (especially for finite x, I know that it can be expanded in power of $\frac{a^{-2}-1}{2}$ cf. first link, probably hypergeometric series?)

Thanks to sound remarks here and here, and looking again at these equations, I noticed that my puzzle boils down to these 2 special functions: $$\int_0^x I_i(u) e^{-a u}du, \quad i=0,1$$ where $I_n(u)$ is the modified Bessel of the first kind.

Are these integral echoing something known in the literature? What can we know for $x<+\infty$ and $\alpha >1$? Especially for finite $x$, I know that it can be expanded in powers of $\frac{a^{-2}-1}{2}$ cf. here, probably hypergeometric series?