# Solving a first order homogenous differential equation

Problem:

Solve the following differential equation.
$$x^2 , dy + (y^2 -xy) , dx = 0$$

$$left( dfrac{x^2}{y^2} right) , dy + left( dfrac{y^2}{x^2} – dfrac{y}{x} right) , dx = 0$$
Hence we have a homogeneous differential equation. Let $$y = vx$$.
begin{align*} dfrac{dy}{dx} &= v + xdfrac{dv}{dx} \ v^2 left( v + xdfrac{dv}{dx} right) &= v^2 – v \ v^3 + v^2 x dfrac{dv}{dx} &= v^2 – v \ v^2 x dfrac{dv}{dx} &= -v^3 + v^2 – v \ x dfrac{dv}{dx} &= -v + 1 – v^{-1} \ dfrac{dv}{-v + 1 – v^{-1}} &= dfrac{dx}{x} end{align*}
$$int dfrac{1}{-v + 1 – v^{-1}} , dv = -dfrac{lnleft(v^2-v+1right)}{2} – dfrac{arctanleft(frac{2v-1}{sqrt{3}}right)}{sqrt{3}}$$
begin{align*} -dfrac{lnleft(v^2-v+1right)}{2} – dfrac{arctanleft(frac{2v-1}{sqrt{3}}right)}{sqrt{3}} &= ln{|x|} + C end{align*}
$$y = dfrac{x}{ln{|x|}+C}$$