I end up with the formula $\sin (d+\frac{(ay+b)}{cx})=y $, and try to write $y$ as a function of $x$. There can be multiple solutions (of $y)$ to $\sin(d+\frac{(ay+b)}{cx})=y$ pretending $x$ is known, but I ONLY interested in the least solutions of $y$. The graph of such $y(x)$ is shown below, however, it doesn't help me to write down its math formula.
I guess such $y(x)$ could be a sum of some series of simple functions. Maybe Fourier transformation is helpful, but I have no clue right now. Any suggestions to attack this problem? Thank you!
I am not sure that I am answering your question; so forgive me if I am off topic.
You cannot explicit $y(x)$ but you can do the reverse and get $$x(y)=-\frac{a y+b}{c \left(d-\sin ^{-1}(y)\right)}$$ So, if $y_n=0$, you can obtain the corresponding $x_n$