Question

Show that the differential equation $\frac{dy}{dx}=\frac{y^2}{xy-x^2}$ is homogeneous and also solve it.

Answer 1

$\frac{dy}{dx}=\frac{y^2}{xy-x^2}=f(x,y)\Rightarrow f(mx,my)=\frac{m^2{y}^2}{m^{2}xy-m^2{x}^2}=\frac{y^2}{xy-x^2}=f(x,y)$

where $m\ne 0.$

Therefore, the differential equation is homogeneous.

$v=\frac{x}{y}\Rightarrow x=vy\Rightarrow \frac{dx}{dy}=y\frac{dv}{dy}+v\frac{dx}{dy}=\frac{xy-x^2}{y^2}=\frac{x}{y}-\frac{x^2}{y^2}\Rightarrow \frac{dx}{dy}=v-v^2=y\frac{dv}{dy}+v\Rightarrow -v^2=y\frac{dv}{dy}\Rightarrow \int \frac{dy}{y}=-\int \frac{dv}{v^2}\Rightarrow \ln y+C=\frac{1}{v}=\frac{y}{x}\Rightarrow y=x(\ln y+C)$