= Solution
This is a <separable differential equation>. Multiply by $e^{-y}$ and integrate:
$$
-e^{-y}=\frac12\ln(1+x^2)+C.
$$
The <initial condition> gives $C=-1$. Hence
$$
\boxed{y(x)=-\ln\left(1-\frac12\ln(1+x^2)\right)}.
$$
For a real solution the outer <logarithm> requires a positive argument. The maximal interval containing the initial point is therefore $\boxed{|x|<\sqrt{e^2-1}}$. At either endpoint $y$ diverges, so the real solution cannot continue through it with a finite value.
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