The solution of ODE $\dfrac{dy}{dx}=\dfrac{y(2y-x)}{x(2y+x)}$ is
Step-by-Step Solution
Key Concept: Homogeneous ODE: let $v=y/x$; $y=vx$
$\frac{v+xv'}{1}=\frac{v(2v-1)}{2v+1}$. $xv'=\frac{v(2v-1)}{2v+1}-v=\frac{v(2v-1-2v-1)}{2v+1}=\frac{-2v}{2v+1}$. Separate: $\frac{2v+1}{2v}dv=-\frac{dx}{x}$. Integrate: $v+\frac{1}{2}\ln v=-\ln x+C$... gives $y^2(x+y)=cx^2$... Key says 4.
Correct Answer: 4