I have an equation of the form $$\frac{x^2}{y} = F(r), $$ where F is a function of $r$. This equation has a solution $r(x,y)$. Suppose we perturb this solution to $r(x + \delta x, y + \delta y)$ for small $\delta x$ and $\delta y$. Can we say anything about how $\delta x$ and $\delta y$ are related to each other?
I tried Taylor expanding this expression and keeping first order terms, but this reduces to $0=0$.
You can differentiate the given equation implicitly, so that $$ \frac{2x}{y}\mathrm{d}x - \frac{x^2}{y^2}\mathrm{d}y = F'(r)\mathrm{d}r = F'(r)\left(\frac{\partial r}{\partial x}\mathrm{d}x + \frac{\partial r}{\partial y}\mathrm{d}y\right) $$ and thus $$ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\frac{x^2}{y^2} + F'(r)\frac{\partial r}{\partial y}}{\frac{2x}{y} - F'(r)\frac{\partial r}{\partial x}} $$