Largest interval of validity for the solution of a first-order DE. I have been trying to solve for the largest interval for which the particular solution of the differential equation
\begin{equation*}
9x^2 \frac{dy}{dx} = y^2+3xy-36x^2, \; \; y(-1) = 0
\end{equation*}
is defined. I've already solved for the particular solution which turned  out to be
\begin{equation*}
\ln|x| + \frac{3}{2\sqrt{5}} \ln \left( \frac{3+\sqrt{5}}{2} \right) = \frac{3}{2\sqrt{5}} \ln \left| \frac{y/x - 3 - 3\sqrt{5}}{y/x - 3 + 3\sqrt{5}} \right|
\end{equation*}
I cannot seem to express this as an explicit function so I can determine the interval of validity. How can I solve for the interval if the particular solution is implicitly defined?

This equation is also a Riccati equation, so you could parametrize $y(x)=-4\dfrac{u(x)}{u'(x)}$ with $u(-1)=0$, $u'(-1)\ne 0$. Inserting this gives
$$
y'=-4+4\frac{uu''}{u'^2}
\\~\\
9x^2y'=-36x^2+36x^2\frac{uu''}{u'^2}
=16\frac{u^2}{u'^2}-12x\frac{u}{u'}-36x^2
\\~\\
\implies 0=9x^2u''+3xu'-4u
$$
This now is an Euler-Cauchy equation with characteristic polynomial
$$
0=9m(m-1)+3m-4=9m^2-6m-4=(3m-1)^2-5
$$
The solution has thus the form
$$
u(x)=A[(-x)^{(\sqrt5+1)/3}-(-x)^{-(\sqrt5-1)/3}]
$$
This is defined on all of $(-\infty,0)$. The derivative in the denominator
$$
u'(x)=\frac{A}3x^{-1}[(\sqrt5+1)(-x)^{(\sqrt5+1)/3}+(\sqrt5-1)(-x)^{-(\sqrt5-1)/3}]
$$
has no roots on that interval.
