I know this is sort of a specific question, but when I was messing around with the equation $xy^{2}=\sin\left(x^{2}+y^{2}+c\right)$ in Desmos (with $c$ being a constant) and I noticed that two seperate lines on the graph intersected and sort of switched parts with each other at about $c=-0.714602$ as well as other c values. I kinda wanted to know what this number was, so I went down a rabbit hole of trying to use calculus to figure it out. I was ultimately unsuccessful. However, I did find that the intersection points were at $(\frac{2}{\sqrt{2}},1)$ and $(\frac{2}{\sqrt{2}},-1)$ by looking at the graph.
So, as a challenge to all you folks out there on the internet, try to find the exact value of $c$ where there are self-intersections in the graph of $xy^{2}=\sin\left(x^{2}+y^{2}+c\right)$ when $-1<c<0$. Also, for bonus points, you can try to figure out why these self-intersections are at $(\frac{2}{\sqrt{2}},1)$ and $(\frac{2}{\sqrt{2}},-1)$.
Helpful tips:
- The correct answer is around $-0.714602$, so your answer should be close to that.
- Looking at the derivative of this equation with respect to $x$ may help, as there will always be some sort of discontinuity (usually $\frac{0}{0}$) at points of self-intersection.
- If you take the derivative with respect to $x$, remember that $\frac{dc}{dx}$ is $0$ because $c$ is a constant.
- There are multiple values of $c$ where the graph has self-intersection points, however once you find one you can find the rest of them.
- This image shows what the graph of the equation should look like with the correct $c$ value:
(cool looking graph, anyways)
Good luck!