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This first order nonlinear differential was posted on a science forum and I am very interested in it. I would like to know what are the most appropriate steps taken to test for the possibility of an analytical solution, as well as what might be the appropriate steps to solve it analytically.

$\dot{x} = - \alpha {x}^{1/2} \arctan{(kx)}$

for $x>0$ and $t \in [0,T]$, where $T < \infty$ and $k > 0$

I'm also kind of curious where it might have come from, which I hadn't asked the original poster and so I do not know! I'm not looking for a lesson, or a complete solution, just an outline if possible. The only book I have on differentials does not cover nonlinear and so I'm not even sure if this is a particular case of some subset of study.

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1 Answer

This is an equation in separeted variables. If $\dot x=dx/dt$, then $$ \frac{dx}{\sqrt{x\,}\arctan(k\,x)}=-\alpha\,dt. $$ Integrating $$ \int\frac{dx}{\sqrt{x\,}\arctan(k\,x)}=-\alpha\,t+C. $$ Unfortunately, I do not think the integral has a closed form in terms of elementary functions (Mathematica can't find it.)

Let $f(x)=-\alpha\,\sqrt{x\,}\arctan(k\,x)$; then $f$ is locally Lipschitz on $[0,\infty)$ (in fact it is differentiable with bounded derivative.) For any $x_0\ge0$, there is a unique solution such that $x(0)=x_0$ defined on a maximal interval $[0,T)$. If $x_0=0$, then $x\equiv0$. If $x_0>0$, it is easy to see that the solution $x$ is global (i.e. $T=\infty$), decreasing, convex, and $\lim_{t\to\infty}x(t)=0$.

For cuatitative results, numerical methods should be used.

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