Question

Solve:$(x^2+3xy+y^2)dx-x^{2}dy=0$

Answer 1

${(x}^2+3xy+y^2)dx-x^{2}dy=0\Rightarrow \frac{dy}{dx}=\frac{{(x}^2+3xy+y^2)}{x^2}⟶(1)$

since the equation is homogeneous differential equation

Let $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$

putting the values in equation (1)

$\Rightarrow x\frac{dv}{dx}+v=\frac{x^2+3x^{2}v+x^2{v}^2}{x^2}\Rightarrow x\frac{dv}{dx}=v^2+2v+1\Rightarrow \frac{dv}{{(v+1)}^2}=\frac{dx}{x}\Rightarrow \int \frac{dv}{{(v+1)}^2}=\int \frac{dx}{x}\Rightarrow -\frac{1}{v+1}+C=\log x\Rightarrow \log x+\frac{1}{\frac{y}{x}+1}=C\Rightarrow \log x+\frac{x}{x+y}=C$