I'm trying to solve the equation, I call it a differential recursive equation, $$y_n(x)=\frac{1}{n}f(x)\frac{dy_{n-1}(x)}{dx},\qquad n=1,2,3,...$$ for $y_n(x)$, where $$y_0(x)=x$$ $y(x)$ and $f(x)$ are differentiable functions of $x$. Is there any general method to solve such recursive and differential equations? Thanks in advance
Edit:
to make it more clear, I assume that $0\le x\le 1$, and $f(x)=a+bx+cx^2+hx^3$.