Is it possible to solve this first order non-linear ODE into a closed form solution (find $r(t)$) ? From first glance it seems to resemble a decay rate equation but has different structure or will I need to resort to numerically solving it.

$$ \frac{dr}{dt} = -K \cdot \Delta R^{n} \cdot \sqrt{r(t)} $$

where $0\leq n \leq 1 $, $K > 0 $ , $\Delta R = r(t) - x(t)$ and given some initial condition $r(0)$. Note that $x(t)$ is known and $r(t)$ is unknown

  • 2
    $\begingroup$ This is a very general kind of equation, there's definitely not a closed form solution for many choices of $n, x(t)$. Numerical is your best bet unless $x(t)$ is constant or $n=0$. $\endgroup$ – AlexanderJ93 Oct 16 '18 at 2:55

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