# First Order Nonlinear ODE - Decay

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

• 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 $n=0$. – AlexanderJ93 Oct 16 '18 at 2:55