I have a doubt respect to resolution of differential equations, for example if we have the family of circles $x^2+y^2=2cx$, deriving $$2x+2y\frac{dy}{dx}=2c$$, combining $$\frac{dy}{dx}=\frac{y^2-x^2}{2xy}$$ and replacing $\frac{dy}{dx}=-\frac{dx}{dy}$, then we have the differential equation $$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}(*)$$ We cannot find the solution of the last differential equation by separation of variables, but if we use polar coordinates in $x^2+y^2=2cx$ we get $$(r\cos\theta)^2+(r\sin\theta)^2=2c(r\cos\theta)$$ and $r=2c\cos\theta$, then $$\frac{dr}{d\theta}=-2c\sin\theta$$ and then $$\frac{r d\theta}{dr}=-\frac{\cos\theta}{\sin\theta}$$
The solution of the last differential equation is $r=2c\sin\theta$, so the solution of the differential equation (*) is $x^2+y^2=2cy$. Then my question is: why the change of coordinates permit find the solution of the differential equation? This is an accident or exit a theorem about this? And if exits such theorem, what kind of differential equation can be solve by change of coordinates? Thanks.