Parametrization of curve

Question

I need to find parametrization of $(1+x)y^2=(y+2x)x$.

I tried to put all $x$-s on one side, and all $y$-s on other but that can't be done.

Could someone please help? Thanks.

Answer 1

Dividing both sides of your equation by $y^2$ [noting that $y=0$ leads to the single solution $(x,y)=(0,0)$] gives $$1+x=(1+2\frac{x}{y})\cdot \frac{x}{y}.$$ This suggests defining $t=x/y$ so that $x=ty.$ Then after algebra the parametrization becomes $$x=2t^2+t-1,\\ y=2t+1-1/t.$$ We never use $t=0$ anyway, since then $x=ty$ gives $x=0$ leading again to the single solution $(x,y)=(0,0).$

Answer 2

Isolating $y$:

$(1+x)y^2=(y+2x)x \Longleftrightarrow y^2+xy^2=xy+2x^2 \Longleftrightarrow y^2+xy^2-xy-2x^2=0$

Now solve the quadratic equation $y^2(1+x)-xy-2x^2=0$ in terms of $y$.

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Do the same thing for isolating $x$:

$-2x^2+xy^2-xy+y^2=0 \Longleftrightarrow -2x^2+x(y^2-y)+y^2=0$

Solve the quadratic equation $-2x^2+x(y^2-y)+y^2=0$ in terms of $x$.

Answer 3

Solving for $y$ (and/or $x$) is only half the battle here. For instance, the quadratic formula applied to this equation $$y^2 (1 + x) - yx - 2 x^2 = 0$$ gives this dual solution for $y$ in terms of "parameter" $x$: $$y = - \frac{x\;(\;-1 \pm \sqrt{9+8 x}\;)}{2\;(1+x)}$$ The "$\pm$" is a bit unsightly, and prevents $y$ from being a function of $x$. We can address this problem by replacing $\sqrt{8x+9}$ with a new parameter, $t$, that we allow to range among positive and negative values.

That is, we take $t^2 = 9+8 x$, so that $$x = \frac{t^2-9}{8}\qquad\text{and}\qquad y = - \frac{t^2-9}{8}\frac{-1+t}{2\left(1+\frac{t^2-9}{8}\right)}= -\frac{t^2-9}{2(t+1)}$$

Note that $t=-1$ is special. The corresponding $x$-value is $-1$; the $y$-value is algebraically undefined right at $t=-1$, but clearly has limits $\pm\infty$ as we approach $-1$ from larger or smaller $t$s: thus, the curve has a vertical asymptote.

Here's an image generated by Mathematica:

The image actually overlays a ContourPlot[] of the original equation with a ParametricPlot[] of the parameterization given. The curves match.