max interval of definition of a Cauchy problem's solution

Question

I have this Cauchy problem : $x(x-1)y'=e^{-y}$ with initial condition $y(2)=\log a,a>0$. I have find the solution :$y(x)=\log(a+\log\frac{2(x-1)}{x})$ But how can I find the max interval of definition?

Answer 1

It will be the largest interval around $2$ on which $y$ is defined. Since the logarithm is defined only for positive numbers, there are two conditions: $$\frac{2(x-1)}{x}>0,\quad a+\log\frac{2(x-1)}{x}>0.$$