Hi how do i solve the following ODE $$(x+y)y'= 3y-x$$given $y(2)=1$.
So here's what I was thinking this is non-linear but I can see it looks like a homogeneous first order ODE so fiddling around I get: $$y'=\frac{3\frac{y}{x}-1}{1+\frac{y}{x}}$$.
But i have no idea how to do go ahead. Any help appreciated.
substituting $$\frac{y}{x}=u$$ then we get $$y=ux$$ and we get $$y'=u'x+u$$ can you finish? and you will get $$\frac{du}{dx}x=\frac{-u^2+2u-1}{u+1}$$ this is separable and now you can write $$-\frac{u+1}{(u-1)^2}du=\frac{dx}{x}$$ does it work now?