# Solutions of a nonlinear trigonometric equation

I have to find the number of solutions of the following equation: $A\sin(x)=\sin(x^2)$ in the interval $x\in[0,k\pi]$ with $A\ge 1$ and $k\in\mathbb{N}$. I don't need the values of the solutions but only how many times the function $f(x)=A\sin(x)$ intersect the $g(x)=\sin(x^2)$. For $A=1$ the solution of the problem is rather simple, but with an arbitrary choice of the parameter $A$ I'm unable to find the number of intersections. Is there some algorithm useful to find them? Thanks in advance.

• I think you might be stuck with numerical methods, other than the trivial solution $x = 0, \forall A$, for example: wolframalpha.com/input/?i=3+sin%28x%29+%3D+sin%28x%5E2%29 . – Amzoti Dec 17 '13 at 17:42
• @Amzoti: I'm looking for an algorithm, but I doubt that there can be one – Riccardo.Alestra Dec 17 '13 at 17:45
• I do not currently see one, other than numerical methods. – Amzoti Dec 17 '13 at 17:47

Since $f(x)=A\sin(x)-\sin(x^2)$ is a continuous equation, you can check the values of $f(x)$ at $x_n=\sqrt{n\pi}$.

If $x_n$ and $x_{n+1}$ have different signs, then there must be at least a root between $x_n$ and $x_{n+1}$. And it is most likely that there is only one root in that range, because when $|x|>|A|$, $f(x)$ is almost monotonic with that range (except for $x$ really close to $x_n$).

The above can be a good approximate if your $|A|\ll k\pi$.

However, to be more accurate,

1) $|x|<\sqrt{\pi/2}$ needs special attention because that part sometimes can have two roots (see second example below).

2) Because $f(x)$ is not monotonic approximately within $|x-x_n|<\sqrt{|A|/x_n}$, you can check the value of $\sqrt{x_n/|A|}|f(x_n)/f'(x_n)|$. If that value is large then it is not likely that there will be additional roots to the first estimate at that range.

Last comment, so in the case of $|A|<1$, the number of roots of $f(x)=0$ within $[0,k\pi]$ can be estimated as $\lceil k^2\pi^2 \rceil$