I need to find $x=x(t)$ and $y=y(t)$ so that the implicitly defined curve on $\mathbb R^2$ $$(2x+y)^2(x+y)=x$$ is converted into an explicit function of the parameter $t$ that can be analysed using single variable calculus.
The usual parametrization $x(t)=t$ and $y(t)=xt$ yielded a very complicated form that was hard to work with. Could you help me with finding a better parametrization? Polar coordinates also didn't help much.
Let $y+2x=u$ therefore $$u^2(u-x)=x\\x=\dfrac{u^3}{1+u^2}$$let $u=t$ therefore $x=\dfrac{t^3}{1+t^2}$ and $$2x+y=t\\\dfrac{2t^3}{1+t^2}+y=t\\y=t\dfrac{1-t^2}{t^2+1}$$and we have$$(x,y)=\left(\dfrac{t^3}{1+t^2},t\dfrac{1-t^2}{t^2+1}\right)$$