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I have the Cauchy problem: $$ \frac{dx}{dt} = \frac{f(t, x)}{g(x) - t^3} \;,\qquad x(t_0) = x_0 \;. $$ It can be shown that when the solution reaches a vicinity of a certain point $(t_\ast, x_\ast)$ the r.h.s. behaves such a way: $$ \frac{dx}{dt} \sim - \frac{\operatorname{sign}(t - t_\ast)}{|t - t_\ast|^{1/2}} \;. $$ This singularity is integrable, so the solution looks like enter image description here

I expect a series of such suingularities in the solution.

The problem is I can't integrate the ODE numerically. Neither explicit (Adams, RK, etc.) nor implicit (BDF) method can pass the singularity. Providing the jacobian doesn't help.

I tried to reformulate the problem, e.g. to define $x(t)$ as a parametric curve on the $(t, x)$ plane, $$ t = t(s) \;,\qquad y(s) = x(t) \;,\qquad (dt/ds)^2 + (dy/ds)^2 \equiv 1 \;. $$ The idea was that $s$ is the natural parametrization so the velocity along the curve is always constant. Doesn't help. The solver can not cope with the fact that $|dt/ds|$ and $|dy/ds|$ may differ from each other by many orders of magnitude.

Reading NumRecipes and such I can't find the proper method also.

UPD: Last news after some break. I couldn't find or construct any numerical method to resolve the caustic. The trick of @LutzL didn't help also (btw, the numerator and the denominator don't turn to zero simultaneously at any time, including caustic).

This problem arised as a self-similar model of a certain gas-dynamic problem. We've made a numerical gas-dynamical solver in a time-space domain and it turned out that a shock wave appears in the soution exactly at the same place the caustic appears (here $t$ is a self-similar variable). This is a kind of a happy end for us. However, the pure algoritmic issue still exists in the self-similar formulation of the problem. I don't know how to solve an ODE with a finite singularity at the r.h.s.

Thanks to everyone who cared.

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    $\begingroup$ What happens if you use a reparametrization that gives you a system \begin{align}\frac{dx}{ds}&=f(t,x),\\\frac{dt}{ds}&=g(x)-t^3?\end{align} Can you give some insight into the geometry of the curves $f(t,x)=0$ and $t^3=g(x)$ and their intersections? $\endgroup$ – LutzL Jun 17 '18 at 8:52
  • $\begingroup$ Hi, @LutzL. Thank you for the trick. I've checked but it doesn't helped. There is another problem. The $f$ and $g-t^3$ are both very large and also differs from each other by many orders. This happened since I actually integrate from right to left, starting from the large $t$. Without your trick the solver is working well until the singularity but in your approach it even can't start since step size becomes too small. I'll try to combine both methods: start with the simple formulation untill the singularity is riched, then pass it with the your trick. $\endgroup$ – Evgeny P. Kurbatov Jun 17 '18 at 10:10

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