I have the following problem: Find the maximal interval for wich the IVP has unique solution. $$\left(x+\dfrac{1}{2}\right)\cos xy^{v}+\sqrt{(\sin x-1)^3}y''+\ln(x+1)y=0$$ $$y^{iv}(\pi)=y'''(\pi)=0,\qquad y''(-\pi)=y'(-\pi)=1,\qquad y(\pi)=2$$
I know that, the unique solution will exist in the interval that contains $\pi$ and $-\pi$ and the functions $f(x)=\left(x+\dfrac{1}{2}\right)\cos x$, $g(x)=\sqrt{(\sin x-1)^3}$ and $h(x)=\ln(x+1)$ are continuos. $f$ doesn't have any problem, but $g$ yes, because we need that $$\sin x-1\geq 0$$ and this happen only for $\dfrac{\pi}{2}+2k\pi$ with $k\in\mathbb{Z}$. To others points $g$ is not defined. And for this, the interval doesn't exists.
Its my answer correct?