So I have the following Cauchy problem :
$$ y' = \frac{e^{-y^2}}{y}$$ $$ y(0) = k \neq 0 $$ I want to find k such that the interval of definition is the biggest possible and containing 0. Now, I started by solving the Differential equation separating variables, and I get the following solution:
$$ \vert y(x) \vert = \sqrt{\log{(2x + e^{k^2})}}$$
However which condition do I impose to find the k such that I get the biggest interval of definition? I only know that I must have: $$ 2x + e^{k^2} > 0$$ $$ 2x + e^{k^2} \ge 1 $$
But after that I am stuck. Can anyone give me a hint?