I'm dealing with a problem that involves solving a pretty nasty differential equation of the shape:
$\frac{\mathrm{d}a}{\mathrm{d}t}=f(a)$
Even though it can be solved through variable separation, the explicit shape of $f(a)$ is so complicated that I cannot obtain a solution. I've exhausted all of my resources on solving the integral with no success; hence, I've come to the conclusion that it does not have a general solution.
Under my advisor's recommendation, I've analyzed the differential equation and obtained the Taylor Series of $f(a)$ both when $a\to 0$ and when $a\to \infty$, which allowed me to obtain the behavior of $\frac{\mathrm{d}a}{\mathrm{d}t}$ at those limits. Since those expressions are simpler, I was able to obtain functions of $a(t)$ under these limits through direct integration. The behaviors of these functions correspond to the ones we were hoping for, so we might be right.
The issue here is that I would love to obtain the full behavior of the function $a(t)$. I'm not writing the explicit expressions of my equations since I'm not looking for the solution, but wondering if there exists a method, a procedure or something that can allow me to extract the full form of $a(t)$ out of its asymptotic behaviors.