Specify the maximum interval for solving the Cauchy problem depending on the initial data

Question

I have a differential equation: $$y'+ \frac{2y}{xln(x)}+ y^2(ln(x))^2=0, x=e, y = y_0$$ I need to find the maximum interval for solving the Cauchy problem depending on the initial data.

So, I found a solution to the differential equation: $$y=\frac{1}{(x+C)(ln(x))^2}$$ How can I find the maximum interval for my case ($x =e, y=y_0)$?

P.S. I don't know what I was doing, but I'll write it down just in case. $$y_0=\frac{1}{C+e}$$ $$C=\frac{1-y_0e}{y_0}$$

Answer 1

Putting $C$ in the solution gives $$y=\frac{1}{(x+C)(\ln(x))^2}=\frac{1}{(x+\frac{1-ey_0}{y_0})(\ln(x))^2}.$$ If $\frac{1-ey_0}{y_0}\ge 0$, then the maximal interval is $(0,\infty)$. If $\frac{1-ey_0}{y_0}< 0$, then the maximal interval is $(0,-\frac{1-ey_0}{y_0})$.