Nonlinear differential equation type Which method should I use for solving equation
$\sqrt{1-x^2}dy + \sqrt{1-y^2}dx = 0$ ?

Use the perfect hints by pedja and  user7530. So dividing both sides by $\sqrt{(1-x^2)(1-y^2)}$ you will be left with something like $$d(\sin^{-1}x+\sin^{-1}y)=0.$$ You may also wish to write the general solution as $$x\sqrt{1-y^2}+y\sqrt{1-x^2}=c$$ or $$x=c\sqrt{1-y^2}-y\sqrt{1-c^2}.$$
