The solution of the differential equation (x^2+y^2) d x-5 x y ~d...

$$\begin{aligned} & \left(x^2+y^2\right) d x=5 x y d y \\ & \Rightarrow \frac{d y}{d x}=\frac{x^2+y^2}{5 x y} \end{aligned}$$
Put $y=V x$$\begin{aligned} & \Rightarrow \mathrm{V}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{1+\mathrm{V}^2}{5 \mathrm{~V}} \\ & \Rightarrow \frac{\mathrm{xdv}}{\mathrm{dx}}=\frac{1-4 \mathrm{~V}^2}{5 \mathrm{~V}} \\ & \Rightarrow \int \frac{\mathrm{V}}{1-4 \mathrm{~V}^2} \mathrm{dV}=\int \frac{\mathrm{dx}}{5 \mathrm{x}} \end{aligned}$<br />Let $1-4 \mathrm{~V}^2=\mathrm{t}\Rightarrow-8 \mathrm{~V} $\mathrm{dV}=\mathrm{dt}$\begin{aligned} & \Rightarrow \int \frac{\mathrm{dt}}{(-8)(\mathrm{t})}=\int \frac{\mathrm{dx}}{5 \mathrm{x}} \\ & \Rightarrow \frac{-1}{8} \ln |\mathrm{t}|=\frac{1}{5} \ln |\mathrm{x}|+\ln \mathrm{C} \\ & \Rightarrow-5 \ln |\mathrm{t}|=8 \ln |\mathrm{x}|+\ln \mathrm{K} \\ & \Rightarrow \ln \mathrm{x}^8+\ln \left|\mathrm{t}^5\right|+\ln \mathrm{K}=0 \\ & \Rightarrow \mathrm{x}^8\left|\mathrm{t}^5\right|=\mathrm{C} \\ & \Rightarrow \mathrm{x}^8\left|1-4 \mathrm{~V}^2\right|^5=\mathrm{C} \\ & \Rightarrow \mathrm{x}^8\left|\frac{\mathrm{x}^2-4 \mathrm{y}^2}{\mathrm{x}^2}\right|^5=\mathrm{C} \\ & \Rightarrow\left|\mathrm{x}^2-4 \mathrm{y}^2\right|^5=\mathrm{Cx} \mathrm{x}^2 \\ & \text{ given } \mathrm{y}(1)=0 \\ & \Rightarrow|1|^5=\mathrm{C} \Rightarrow \mathrm{C}=1 \\ & \Rightarrow\left|\mathrm{x}^2-4 \mathrm{y}^2\right|^5=\mathrm{x}^2\end{aligned}$$$