Does every two variable equation are separable? I am a first year undergraduate student(Not undergone group theory yet) and i describe my situation as follows:
I was trying to trace curve of equation say :
$$x(y-x)^2=ay^2$$
So as we are taught in school by simple steps to work like :
(1) Symmetry

(2) Origin & Tangent at origin

(3) Point of Intersection with axes.

(4) Asymptotes

(5) Region of absence.

(6) Some optimization techniques etc.

I found the (5) as very helpful if you can express y in terms of then you have a very firm idea about its nature.

Say i have an equation :
$$2x^2-2xy+y^2-4x+2y+1=0$$
So at first look it seems to not expressed in terms of $y=f(x)$
But it is actually ;
$$y=(x-1)\pm(2x-x^2)^\frac{1}{2}$$

QUESTION
Is it possible to express every equation in the form of $y=f(x)$ say of degree 2,3,4 and 5 ?
Can you express these ?
$$x(y-x)^2=ay^2$$
$$a^5y^2+2a^3x^3y+x^7=0$$
I guess for these type of equation say more than 5 i must use polar co-ordinates.
Please let me know what i can do in such situation to trace its graph ?
Thanks

In general no, because to put an equation in the form y=f(x) you need for there to only be one y value for each x value.
You can cheat a little on this, as you have in one example, by using ±, but this won't work in general. For example, it has been proven that there is no closed form solution to 5th order (and beyond) polynomials, so most equations involving $y^5$ and beyond can't be expressed this way at all (in the sense that such an expression doesn't even exist, not just that you can't find it reliably), even if you did allow multi-value operators in your equation.
That said, your two examples are quadratic in y, and so they can be solved. Hint: express them in the form $ay^2+by+c=0$, where 'a', 'b' and 'c' may each be functions of x, then use the quadratic formula to get:
$y = \frac{-b±\sqrt{b^2-4ac}}{2a}$
If you want to plot functions you can't write as y = f(x), you could try writing x = f(y), but that will only work sometimes. Any other tricks I can imagine would be specific to particular equations. Personally, at some point I would just stop trying to do it by hand and use a suitable software tool.
