Case Study-III
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\frac{\mathrm{dy}}{\mathrm{d}\mathit{x}}=\mathrm{F}\left(\mathit{x},\mathrm{y}\right)$ is said to be homogeneous if $\mathrm{F}\left(\mathit{x},\mathrm{y}\right)$ is a homogeneous function of degree zero, whereas a function $\mathrm{F}\left(\mathit{x},\mathrm{y}\right)$ is a homogenous function of degree $\mathrm{n}$ if $\mathrm{F}\left(\mathit{\lambda }\mathit{x},\mathit{\lambda }\mathrm{y}\right)={\mathit{\lambda }}^{\mathrm{n}}\mathrm{F}\left(\mathit{x},\mathrm{y}\right)$. To solve a homogeneous differential equation of the type $\frac{\mathrm{dy}}{\mathrm{d}\mathit{x}}=\mathrm{F}\left(\mathit{x},\mathrm{y}\right)=$ $\mathrm{g}\left(\frac{\mathrm{y}}{\mathit{x}}\right)$, we make the substitution $\mathrm{y}=\mathrm{v}\mathit{x}$ and then separate the variables. Based on the above, answer the following questions :
(I) Show that $\left({\mathit{x}}^2-{\mathit{y}}^2\right)\mathrm{d}\mathit{x}+2\mathit{x}\mathit{y}\mathit{d}\mathit{y}=0$ is a differential equation of the type $\frac{\mathrm{dy}}{\mathrm{d}\mathit{x}}=\mathrm{g}\left(\frac{\mathrm{y}}{\mathit{x}}\right)$.
(II) Solve the above equation to find its general solution.