I have been trying to solve a differential equation using the Laplace transform. I don't provide that expression here since it is quite too large IMO, and rather to asking for the solution, I am curious about some of the terms, whose Laplace transform are, apparently, the same.
What we know, from Integrals and Series: Direct Laplace transforms, Volume n. 4, Prudnikov et al: \begin{align} \mathcal{L}\{f(t)\} &= F(s) = \int_0^\infty e^{-st} f(t) \, dt \tag{1} \\ \mathcal{L}\left\{\frac{f(t)}{t^2}\right\} &= \int_s^\infty \int_s^\infty F(s) (ds)^{2}=\int_s^\infty \frac{(u-s)}{1!} F(u) \, du \tag{2} \\ \mathcal{L}\{t^2 f(t)\} &= \frac{\partial^2}{\partial s^2}F(s) \tag{3} \end{align}
So, the terms thar appear in my differential equation are the following: $$\mathcal{L}\left\{\frac{f(t)}{t^2} \right\}=\int_s^\infty \int_s^\infty F(s) (ds)^{2}=\int_s^\infty \frac{(u-s)}{1!} F(u) \, du \tag{4} $$
Let $q=s+\alpha$
\begin{align} \mathcal{L}\left\{\frac{e^{-\alpha t } f(t)}{t^2}\right\} &= \int_0^\infty e^{-(s+\alpha)t} \frac{f(t)}{t^2} \, dt = \int_0^\infty \frac{e^{-qt} \,f(t)}{t^2} \, dt \\ &= \int_s^\infty \frac{(u-s-\alpha)}{1!} F(u) \, du \tag{5} \end{align}
And let $p=s-\alpha$ \begin{align} \mathcal{L}\left\{\frac{e^{\alpha t}f(t)}{t^2} \right\} &= \int_0^\infty e^{-(s-\alpha)t} \frac{f(t)}{t^2} \, dt = \int_0^\infty \frac{e^{-pt} \, f(t)}{t^2} \, dt \\ &= \int_s^\infty \frac{(u-s+\alpha)}{1!} F(u) \, du. \tag{6} \end{align} I do not know if this way of seeing the Laplace transforms may be correct, or not. However, if we suppose is correct (Is it?) and then we arrive at some expression like this one: \begin{align} \int_s^\infty \frac{(u-s+\alpha)}{1!} F(u) \, du &+ \int_s^\infty \frac{(u-s-\alpha)}{1!} F(u) \, du + \int_s^\infty \frac{(u-s)}{1!} F(u) \, du \\ & \hspace{5mm} + F(s+\alpha)+F(s-\alpha)+sF(s) = 0 \end{align}
The $F(s)$, $F(s+\alpha)$ and $F(s-\alpha)$ came from other terms on the original differential equation.
By the way, How it is possible to use this curious $F(u)$ terms that are more likely a dummy variable, to solve my ODE?
Have you seen something like this before, or do you know some book that may explain a little bit of usages on these dummy integrals that may appear in the Laplace transforms.
Thanks in advance