# Numerical scheme to solve ODE with singularity

I've been struggling with the following. I'd like to find a numerical scheme to solve the following ODE:

$$x'' + \frac{1}{x^2} = 0$$

Most schemes work quite well when $$x$$ isn't too close to zero, but I need one that is robust to traversing the origin. E.g., I expect that if my initial conditions are $$(x, x') = (1, 0)$$, the solution would be a kind of oscillator between $$1$$ and $$-1$$.

I'm not sure this is even possible to find? I've done some research and tried to find a solution myself, but I'm hitting a wall. Does anyone have some insights on this problem?

Thanks for your help!

## 2 Answers

Before you look for a numerical scheme, ask whether there is a solution at all. The differential equation is undefined at $$x=0$$, so there is really no such thing as "traversing the origin" for its solutions. In fact, since $$x'' < 0$$ everywhere, it is impossible to continue a solution after it hits the origin with velocity $$-\infty$$, unless you make the velocity jump to $$+\infty$$. You could, for example, bounce off the origin (so $$x(t_0+t) = x(t_0-t)$$ where $$t_0$$ is a time you hit the origin). But if you somehow made it to some $$x < 0$$ with $$x' < 0$$, $$x'$$ will stay negative so you can never go back.

EDIT: In fact, by "conservation of energy", $$E(x,v) = \frac{v^2}{2} - \frac{1}{x}$$ is conserved, where $$v = dx/dt$$ is the velocity.
Trajectories in the $$(x,v)$$ plane look like this. For positive $$x$$, solutions can come from $$x \to 0, v \to +\infty$$ at some finite time in the past and go back with $$x \to 0, v \to -\infty$$ at some finite time in the future, or come from $$(0, +\infty)$$ and go to $$x \to +\infty$$ as $$t \to +\infty$$, or come from $$x \to +\infty$$ as $$t \to -\infty$$ and go to $$(0,-\infty)$$. For negative $$x$$, solutions must come from $$x \to -\infty, v \to const > 0$$ as $$t \to -\infty$$ and go back to $$x \to -\infty$$, $$v \to -const < 0$$ as $$t \to +\infty$$ without ever approaching $$x=0$$.

Before comment for numerical solution it is of interest to see to what leads the analytic solution : $$\frac{d^2x}{dt^2}+\frac{1}{x^2}=0$$ $$2\frac{d^2x}{dt^2}\frac{dx}{dt}+\frac{2}{x^2}\frac{dx}{dt}=0$$ $$\left(\frac{dx}{dt}\right)^2-\frac{2}{x}=c$$ With initial condition $$(x,x')=(1,0)$$ at $$t=0$$ , equivalently $$\begin{cases} x(0)=1 \\ x'(0)=0\end{cases}$$ $$0^2-\frac21=c\quad\implies\quad c=-2$$ $$\left(\frac{dx}{dt}\right)^2=\frac{2}{x}-2\quad\implies\quad 0\leq x< 1$$ $$\frac{dx}{dt}=\pm\sqrt{\frac{2}{x}-2}$$ $$dt=\pm\sqrt{\frac{x}{2(1-x)}}dx\qquad 0\leq x< 1$$

Let $$x(t)=\cos^2(\alpha(t))$$ $$dt=\pm \sqrt{2}\cos^2(\alpha)d\alpha$$ $$t=\pm \sqrt{2}\int\cos^2(\alpha)d\alpha$$ $$t=\pm\frac{1}{\sqrt{2}}\big(\alpha+\sin(\alpha)\cos(\alpha) \big)+\text{constant}$$ The constant is eliminated with condition $$x(0)=1$$ then $$\alpha(0)=0$$.

The analytic solution expressed on parametric form is : $$\boxed{\begin{cases} t=\pm\frac{1}{\sqrt{2}}\big(\alpha+\sin(\alpha)\cos(\alpha) \big)\\ x=\cos^2(\alpha) \end{cases}}$$ One cannot express $$x(t)$$ on closed form because the equation $$\quad t(x)=\pm\frac{1}{\sqrt{2}}\left(\cos^{-1}(\sqrt{x})+\sqrt{x(1-x)} \right)\quad$$ cannot be inverted on the form of a finite number of elementary functions.

The above parametric form is the simplest to study the solution, which clearly is periodic. (Period$$=\pi\sqrt{2}$$ ). COMMENT about numerical solution.

I suppose that the starting point is for $$t=0$$ with $$x=1$$ and $$x'=0$$ then $$x''=-\frac{1}{1^2}=-1$$.

There is no difficulty with successive small increments of $$t$$ to compute the successive values of $$x''$$ then $$x'$$ and next $$x$$.

The difficulty arrises when $$t\simeq\frac{\pi}{\sqrt{2}}$$ corresponding to $$x\simeq 0$$ because $$x''$$ becomes big. A possible way to overcome the difficulty is to change the origin of $$t$$ and restart the process backwards from another point, for example $$t=\pi\sqrt{2}$$, then forwards up to $$t\simeq 3\frac{\pi}{\sqrt{2}}$$, and so on.

But definitively the simplest method is to use the parametric equation and draw $$\big(t(\alpha)\:,\:x(\alpha)\big)$$ for $$\alpha$$ from $$0$$ to any large value of $$\alpha$$.