Strictly speaking, the differential equation isn't defined at $x=0$. Thus the maximal domain on which you can consider this is $\mathbb R\setminus\{0\}$, and there's nothing to constrain the solutions on both sides of $0$ to have anything to do with each other; thus you can choose their parameters $a$ and $b$ independently of each other. You can also use $c_+\mathrm e^x+c_-\mathrm e^{-x}$ if you like, by the way.
Your guess $y(x)=b\sinh x$ is correct, except again you can choose different parameters $b$ to the left and right of $0$. For $x\to\infty$, the exponential representation above is more conducive for seeing the answer immediately.