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 $$\frac{d y}{d x}=F(x, y)$$ is said to be homogeneous if $$\mathrm{F}(x, y)$$ is a homogeneous function of degree zero, whereas a function $$\mathrm{F}(x, y)$$ is a homogenous function of degree $$n$$ if $$\mathrm{F}(\lambda x, \lambda y)=\lambda^n F(x, y)$$. To solve a homogeneous differential equation of the type $$\frac{d y}{d x}=\mathrm{F}(x, y)= g\left(\frac{y}{x}\right)$$, we make the substitution $$y=v x$$ and then separate the variables.

Based on the above, answer the following questions:

(i) Show that $$\left(x^2-y^2\right) d x+2 x y d x=0$$ is a differential equation of the type $$\frac{d y}{d x}=g\left(\frac{y}{x}\right)$$.

(ii) Solve the above equation to find its general solution.