<s>[INST] Solve the first order differential equation, $\frac{x\mathop{dx}-y\mathop{dy}}{x\mathop{dy}-y\mathop{dx}}=\sqrt{\frac{1+x^2-y^2}{x^2-y^2}}$ I need to find the general solution to the differential equation,
$$\dfrac{x\mathop{dx}-y\mathop{dy}}{x\mathop{dy}-y\mathop{dx}}=\sqrt{\dfrac{1+x^2-y^2}{x^2-y^2}}$$
I have tried simplifying it upto
$$\mathop{d\sqrt{1+x^2-y^2}}=\dfrac{x\mathop{dy}-y\mathop{dx}}{\sqrt{x^2-y^2}}$$
but but do not know what to do with the RHS. How to solve the differential equation?
 [/INST] Substitute $x=r \sec \theta$, $y=r \tan \theta$. 
$x^2 - y^2 = r^2$
Taking differentials, $xdx - y dy = r dr$ 
and $\frac y x=\sin\theta$
Taking differentials, $\dfrac {x dy - y dx} {x^2} = \cos \theta d\theta$
or, $x dy - y dx = r^2 \sec \theta d\theta$
Thus the differential equation becomes
$\dfrac {rdr}{r^2 \sec\theta d\theta} =\sqrt{\dfrac{1+r^2}{r^2}}$
Can you follow from here?
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