So I'm trying to numerically solve a second order ODE of the form:
$\frac{d^2x}{dt^2} = nx + \frac{C}{t}$
But there are two… complicated boundary conditions.
$\lim_{t\to0} = 0$
$\lim_{t\to\infty} = 0$
I've set up some code for RK4. Essentially, I've been setting my initial values as $t = 0.000001$, $x=0.000001$ then evaluating up to some really large number and checking if that's close enough to 0. Then I've been testing random combinations of $\frac{dx}{dt}\rvert_{t=0}$ and $n$, but I'm not sure that this is the right approach.
What is typically done for these sorts of conditions?