Something like this?
It's made in 3d, with a macro for the coils, and it needs layers to fix the visibility and a couple of tikz styles (arrows, and borders in some lines).
A PSTricks answer for the PSTricks question.
ODEs can be numerically solved with package pst-ode (RKF45 method).
The given equation is solved separately for the lower and for the upper half-spaces using initial conditions (1e-9, 0, -1) and (1e-9,...
Here is how to do it easily with pst-plot. You just have to plot the function x² twice, once with a solid line, one with a dashed line. Note it can perfectly be compiled with pdflatex if you load auto-...
If you slightly increase the maximum value of the y coordinate in \psaxes, it's OK. Maybe this comes from a rounding error.
The best way to explain is showing the animations, right?
With curve plotstyle
It needs at least 3 points. When N=2 there is no graph.