We have a function:

\begin{equation} |x|+|y|\frac{dy}{dx} = \sin \left(\frac{x}{n}\right) \end{equation}

We are to attempt to express the solution analytically and also approximate them. To develop a basic approximate solution, we will leverage a recursive simulation approach. We first set a constant N which in the N value which we will eventually vary. N = 0.5 We can get some values by stepping through x and y through which we can then figure \frac{dy}{dx}, namely, how the function evolves.

cache res = [] # number of steps steps = 1000 # seed values x = -5 y = 5 # step size step = 1/100 # for number of setps for _ in range(steps): # get the current equation and slope solution dydx = (sin(x/N)-abs(x))/abs(y) # append result res.append((x,y,dydx)) # apply the slope solution to iterate next y # step size is defined by `step` x += step y += dydx*step We have now a set of analytic solutions (x,y,\frac{dy}{dx}). Let’s plot them!

scatter_plot([i[0:2] for i in res]) Great, now we have a fairly non-specific but “correct” solution. We are now going to try to derive an analytic solution. Wait… That’s not the solution we got! But… its close: the blue line simply need to be reflected across the x axis. Its actually fairly apparent why we will need this negative. We just declared that y was negative for that portion of the solution; the output of a square root could never be negative, so of course to achieve y being negative we have to take into account that square roots have a possible negative output as well. Nice; now our analytical results agree with out numerical results.

\begin{equation} \begin{cases} y>0 & y=\sqrt{-2n\cos\left(\frac{x}{n}\right)-x\vert x\vert} +C \\ y<0 & y=-\sqrt{2n\cos\left(\frac{x}{n}\right)+x\vert x\vert}+C \end{cases} \end{equation}

Moving on to the result of the questions. Solution behavior The solution are unbounded and mostly decreasing. As n\in [-1,1], the solution becomes unstable; a solution does not exist at n=0. At n=0.5, a solution passes through (0,-1).