I want to know if can be find the conditions for these 3 constants, lets named them "$a$", "$b$", and "$c$", that makes to be finite the following integral:
Let be $y(t)$ an arbitrary continuous function that is described by the following differential equation: $$ y'' +a\,y'+b\,y+c = 0,\,\, y(t_i),\,\,y'(t_i)$$
And, also let extract a time-finite sample of $y(t)$ between times ${t_0,\,t_F}$, with border conditions $y(t_0)$, $y(t_f)$, $y'(t_0)$, and $y'(t_f)$.
I want to know if is possible to find the values of "$a$", "$b$", and "$c$" so this happen to be true (and also, what are the "threshold" values of them):
$$ \int\limits_{-\infty}^\infty \left| \frac{e^{-iwt_F}\left(y'(t_f)+y(t_f)(iw+a)+\displaystyle{\frac{ic}{w}} \right)-e^{-iwt_0}\left(y'(t_0)+y(t_0)(iw+a)+\displaystyle{\frac{ic}{w}}\right)}{w^2-iwa - b}\right| \,dw < \infty $$
Please explain also how you get your results. Beforehand, thanks you very much.
Added later:
Here what I have tried: first I move the term $\frac{1}{w}$ out of the numerator: $$ \left| \frac{e^{-iwt_F}\left(w y'(t_f)+y(t_f)(iw^2+a w)+ic \right)-e^{-iwt_0}\left( w y'(t_0)+y(t_0)(iw^2+a w)+ic\right)}{w (w^2-iwa - b)} \right| $$
Then, by finding the roots of the denominator, I will have it goes to zero (so the integral is going to infinity), when: $$w_1 = 0, \,\,w_{2,\,3} = \frac{i}{2}\cdot(a\pm \sqrt{a^2-4 b})$$
But since when $w \to 0$ the numerator is also zero, and it haves exponential terms dependent of $w$, bu L'Hopital I think $w_1 = 0$ is not a solution that makes the fraction diverges.
Also, since $w$ is the angular frequency and its defined here to be $w \in \mathbb{R}$, for the other $w_2$ and $w_3$ they will be complex for any $a \neq 0$ so if the $y'$ term is present, the fraction will be finite, and for the case that $a=0$, the fraction will diverge when $w_{2,\,3} = \pm \sqrt{b}$.
But the divergence of the fraction is not enough to assure the convergence of the integral.
If I have that $y(t_0)=y(t_f)=y'(t_0)=y'(t_f)=0$, then I will be integrating something of the form: $$ \int\limits_{-\infty}^\infty \left|\frac{\text{Bounded}}{\mathbb{P}(w^3)+\text{non-zero constant}}\right| dw $$ with $\mathbb{P}(w^3)$ a polynomial of order 3 with positive first $w^3$ term, so it will be finite.
But I don't know how to check for the convergence of integrals for any other case, so any help of how to do it or explain the remaining cases will be useful.
Also, by experimenting in Wolfram-Alpha, I believe that this is happening: $$ \int\limits_{-\infty}^{\infty} \left| \frac{1}{w^2+ib} \right| dw < \infty,\,\,\,\forall b\in \mathbb{R} $$
But, $$ \int\limits_{-\infty}^{\infty} \left| \frac{1}{w^2+b} \right| dw \begin{cases} = \infty,\,\,\, b \leq 0 \\ < \infty,\,\,\, b > 0 \end{cases}$$
So the complex unit becomes harder to form an intuition related to this, I remember that the Laplace transform have this kind of issues also, but I don't know how to relate them.
