Transcendental equation $x - c \sin(x)=0$

Question

Hy, well I have a problem with the transcendental equation $x - c \sin(x) = 0$, where $c$ is some positive constant. I tried using Newton's method for finding the roots but it didn't work well. The problem is also the number of solutions, because as $c$ is increasing, there will be more solutions.

Answer 1

Let's look at the two functions $y=x$ and $y=c\sin(x)$ and see where they intersect.

Depending on the constant $c \gt 0$, we could have a different number of solutions. With a very large $c$, the graph will have a large amplitude and oscillate across the line many times, while a very small $c$ close to $0$ will cause $c\sin(x)$ to appear identical to the line, but not quite be the same line. To my knowledge there isn't a simple formula for the solutions using an arbitrary $c$. However, from observation we can see $x=0$ is always a solution. (Why?)

I would suggest picking a value for $c$ and then going from there.