The general solution of $y''=-\sin y$ When I asked Mathematica to solve the ODE
$$y''=-\sin y \tag{1} $$
I got the solutions
$$y=\pm 2 \text{am}\left(\frac{1}{2} \sqrt{\left(c_1+2\right) \left(t+c_2\right){}^2}|\frac{4}{c_1+2}\right), \tag{2} $$
where $\text{am}(u|m)$ is the Jacobi Amplitude function. I wonder why there is a $\pm$ ambiguity here (I believe it's related to the square root), since the equation $(1)$ is explicit. 
P.S.
Using the rules $c_1 \mapsto -2+4c_1^2,c_2 \mapsto c_2/c_1$ one gets the equivalent form
$$y= \pm2 \text{am}\left(\sqrt{\left(t c_1+c_2\right){}^2}|\frac{1}{c_1^2}\right)$$
and if one cancels the square root with the square, noticing that am is odd in the first argument one gets an even nicer form
$$y=\pm2 \text{am}\left(c_1t +c_2|\frac{1}{c_1^2}\right).$$
However, my question still remains: Why is there an ugly $\pm$ in the solution if the ODE is written explicitly in the form $y''=f(x,y,y')$? Is there a nicer form for the general solution?
Thank you!
 A: $$y=\pm2 \text{am}\left(c_1t +c_2\bigg|\frac{1}{c_1^2}\right).$$
The Jacobi amplitude function is symmetrical : $\quad\text{am}(-x|k)=-\text{am}(x|k)$ 
$$2\text{am}\left(c_1t +c_2\bigg|\frac{1}{c_1^2}\right)=-2\text{am}\left(-(c_1t +c_2)\bigg|\frac{1}{c_1^2}\right) = -2\text{am}\left(C_1t +C_2\bigg|\frac{1}{C_1^2}\right)$$ 
with $C_1=-c_1$ and $C_2=-c_2 .$
This means that, considering the general solution of the ODE one can forget the $\pm\:$: 
$$y=\pm2 \text{am}\left(c_1t +c_2\bigg|\frac{1}{c_1^2}\right) \equiv  2\text{am}\left(c_1t +c_2\bigg|\frac{1}{c_1^2}\right) \text{insofar } c_1,c_2 \text{ are any constants.}$$
Of course, this isn't true if we don't consider the whole set of solutions, but one particular solution, according to initial/boundary conditions : The conditions determine the constants and the sign. That is why it is better to write :
$$y=\pm2 \text{am}\left(c_1t +c_2\bigg|\frac{1}{c_1^2}\right)\:,$$
knowing that, for the particular solution,  $\pm$ means $+$ or $-$ (i.e.: only one solution, not both).
NOTE : 
The question itself is somehow ambiguous : " I wonder why there is a $\pm$ ambiguity here , since the equation $(1)$ is explicit ". 
It isn't specified if the subject of the question concerns an ODE alone, or an ODE with initial/boundary conditions.
Not only in the case of equation $(1)$, but in all cases of ODEs : If the question concerns an ODE without initial/boundary conditions  "explicit" doesn't mean that only one solution exist. They are an infinity of solutions since the constants such as $c_1$, $c_2$, (and if they are $\pm$ in it), all can be chosen among many values. One cannot say that there is an ambiguity due to the presence of arbitrary constants and $\pm$ in the general solution. 
If the question concerns an ODE with initial/boundary conditions, insofar the conditions are consistent for a unique solution, those conditions determines the values of the constants and the signe affected to $\pm$.
