The solution of the differential equation (dy)/(dx)=-(x^2+3y^23x^2+y^2),y(1)=0 is :

Question

The solution of the differential equation

$\frac{dy}{dx}=-(\frac{{x}^{2}+3{y}^{2}}{3{x}^{2}+{y}^{2}}),y(1)=0$ is :

Accepted Answer

$y=vx$

$v+x\frac{dv}{dx}=-(\frac{1+3{v}^{2}}{3+{v}^{2}})$

$x\frac{dv}{dx}=-(\frac{1+3{v}^{2}}{3+{v}^{2}}+v)$

$\frac{dv}{dx}=-(\frac{{(1+v)}^{3}}{3+{v}^{2}})$

$\Rightarrow \frac{3+{v}^{2}}{{(1+v)}^{3}}=\frac{-dx}{x}$

$\Rightarrow \ln |v+1|+\frac{2v}{{(v+1)}^{2}}=C-\ln |x|$

$x=1,v=0\Rightarrow C=0$

$\Rightarrow \ln |x+y|-\ln |x|+\frac{2xy}{{(x+y)}^{2}}=-\ln |x|$

Solutions · 1