The general solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y+\sqrt{x^2-y^2}}{x}\) is

\(\begin{aligned}
& \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y+\sqrt{x^2-y^2}}{x} ...(i)\\
& \text { Put } y=\mathrm{v} x ...(ii)\\
\therefore \quad & \frac{\mathrm{~d} y}{\mathrm{~d} x}=\mathrm{v}+x \frac{\mathrm{~d} v}{\mathrm{~d} x}
...(iii)\end{aligned}\)
Substituting (ii) and (iii) in (i), we get
\(\begin{array}{ll} & \mathrm{v}+x \frac{\mathrm{dv}}{\mathrm{d} x}=\frac{\mathrm{v} x+\sqrt{x^2-\mathrm{v}^2 x^2}}{x} \\ \therefore & \mathrm{v}+x \frac{\mathrm{~d} v}{\mathrm{~d} x}=\mathrm{v}+\sqrt{1-\mathrm{v}^2} \\ \therefore & x \frac{\mathrm{dv}}{\mathrm{d} x}=\sqrt{1-\mathrm{v}^2} \\ \therefore & \int \frac{\mathrm{dv}}{\sqrt{1-\mathrm{v}^2}}=\int \frac{\mathrm{d} x}{x} \\ \therefore & \sin ^{-1}(\mathrm{v})=\log |x|+\mathrm{c} \\ \therefore & \sin ^{-1}\left(\frac{y}{x}\right)=\log |x|+\mathrm{c}\end{array}\)