I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution?

$$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 (\frac{\partial y}{\partial x})} \partial_y\right) (\partial_y f(y(x)))^2 =b^2$$ or in a cleaner way: $$ \left(1-w(y)\partial_y\right)f(y)-1/4\left(1-\frac{1}{w(y)}\partial_y \right) \left(\partial_y f(y)\right)^2=b^2 $$ with $$ w(y)=x^3\frac{\partial y}{\partial x}, \ \ y=y(x), \ \ b=const. $$

--Edited from comments--- This question comes from a problem in classical general relativity, involving an Ad$S_5$ background metric. The differential equation comes from trying to embed a string in this background. The function $f$ is related to the shape of the string, whereas $y$ and $x$ are related to the worldsheet coordinates. The unknowns are both $f$, and $y$.