Consider the ode 
$$
f''(t)-e^{-2t} f(t)=0.
$$
What is the general behaviour of $|f|$ for large $t$s? 


Is it true that there exists an $A\in \mathbb{R}$ and there exists a positive $B$ such that $|f(t)|\sim A+Bt $ for all large enough $t$?



Asymptotics of solutions to the ODE $f''(t)-e^{-2t} f(t)=0$
The general solution of this differential equation is
$$ f \left( t \right) =a\,{{ I}\_{0}\left({{\rm e}^{-t}}\right)
}+b\,{{ K}\_{0}\left({{\rm e}^{-t}}\right)}
$$
where $I\_0$ and $K\_0$ are modified Bessel functions.
As $t \to +\infty$,
$$I\_0({\rm e}^{-t}) = 1 + O\left({\rm e}^{-2t}\right)$$ while
$$K\_0({\rm e}^{-t}) = \ln(2) - \gamma + t + O\left({\rm e}^{-2t}\right) $$


