Find the general solution of the differential equation $\displaystyle \frac{d \mathrm{y}}{d \mathrm{x}}=\frac{\mathrm{x}^{2}+\mathrm{y}^{2}}{2 \mathrm{x} \mathrm{y}}$.

$$\begin{aligned} \text { Put } \mathrm{y} & =\mathrm{vx} \text { so that } \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}} \\ & \mathrm{v}+\mathrm{x} \frac{d \mathrm{v}}{d \mathrm{x}}=\frac{\mathrm{x}^{2}+\mathrm{v}^{2} \mathrm{x}^{2}}{2 \mathrm{x} \mathrm{v} \mathrm{x}} \\ & \int \frac{1}{\mathrm{x}} d \mathrm{x}=\int \frac{2 \mathrm{v}}{1-\mathrm{v}^{2}} d \mathrm{v} \\ & \Rightarrow \log |\mathrm{x}|=-\log \left|1-\mathrm{v}^{2}\right|+\log \mathrm{C} \\ & \log \left|\mathrm{x}\left(1-\mathrm{v}^{2}\right)\right|=\log \mathrm{C} \\ & \Rightarrow \mathrm{x}\left(1-\frac{\mathrm{y}^{2}}{\mathrm{x}^{2}}\right)=\mathrm{C} \text { or } \mathrm{x}^{2}-\mathrm{y}^{2}=\mathrm{C} \mathrm{x} \end{aligned} $$