How Do I generate a Solution equation of this condition? **Question** : Solution of the differential equation $$x=1+xy\frac{dy}{dx}+\frac{x^2y^2}{2!}(\frac{dy}{dx})^2+\frac{x^3y^3}{3!}(\frac{dy}{dx})^3+\dot{}\dot{}\dot{}\dot{}\dot{}$$ Is? **Attempt** : My first thought went to Taylor's Expansion and tried to find a function which follows $$f(x)=f(0)+\frac{x}{1!}f^{'}(0)+\frac{x^2}{2!}f^{"}(0)+\frac{x^3}{3!}f^{'''}(0)+\dot{}\dot{}\dot{}\dot{}\dot{}$$ My initial guesses frames out that it must be something like $y=\ln x$ or $xy=x^y$ type function nut i can't pin-point any with a convincing method. Any suggestions or procedure will be appreciated. Cheers!

The series is the exponential function with the arguement $xy \frac{dy}{dx}$ so the equation is \begin{eqnarray*} x= e^{ xy \frac{dy}{dx}} \end{eqnarray*} Take logarithms & integrate ... we have \begin{eqnarray*} y^2= (ln x)^2 +C \end{eqnarray*}