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(x2+y2–1)3-x2y3=0

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(x2+y2–1)3-x2y3=0 equation

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The solution set is obviously symmetric with respect to the yy-axis. Therefore we may assume x0x\geq 0. In the domain {(x,y)R2  x0}\{(x,y)\in {\mathbb R}^2\ |\ x\geq0\} the equation is equivalent with\nx2+y21=x2/3y ,x^2+ y^2 -1=x^{2/3} y\ ,\nwhich can easily be solved for yy:\ny=12(x2/3±x4/3+4(1x2)) .y={1\over2}\bigl(x^{2/3}\pm\sqrt{x^{4/3}+4(1-x^2)}\bigr)\ .\nNow plot this, taking both branches of the square root into account. You might have to numerically solve the equation x4/3+4(1x2)=0x^{4/3}+4(1-x^2)=0 in order to get the exact xx-interval.

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Numerical solution:

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The solution

You have entered [src]
(x2 + y2 - 1)*3 - x2*y3 = 0
x2y3+(x2+y21)3=0- x_{2} y_{3} + \left(x_{2} + y_{2} - 1\right) 3 = 0
Detail solution
Given the linear equation:
(x2+y2-1)*3-x2*y3 = 0

Expand brackets in the left part
x2*3+y2*3-1*3-x2*y3 = 0

Looking for similar summands in the left part:
-3 + 3*x2 + 3*y2 - x2*y3 = 0

Move free summands (without y3)
from left part to right part, we given:
x2y3+3x2+3y2=3- x_{2} y_{3} + 3 x_{2} + 3 y_{2} = 3
Move the summands with the other variables
from left part to right part, we given:
x2y3+3y2=3x2+3- x_{2} y_{3} + 3 y_{2} = - 3 x_{2} + 3
Divide both parts of the equation by (3*y2 - x2*y3)/y3
y3 = 3 - 3*x2 / ((3*y2 - x2*y3)/y3)

We get the answer: y3 = 3*(-1 + x2 + y2)/x2
The solution of the parametric equation
Given the equation with a parameter:
x2y3+3x2+3y23=0- x_{2} y_{3} + 3 x_{2} + 3 y_{2} - 3 = 0
The coefficient at y3 is equal to
x2- x_{2}
then possible cases for x2 :
x2<0x_{2} < 0
x2=0x_{2} = 0
Consider all cases in more detail:
With
x2<0x_{2} < 0
the equation
3y2+y36=03 y_{2} + y_{3} - 6 = 0
its solution
y3=3y2+6y_{3} = - 3 y_{2} + 6
With
x2=0x_{2} = 0
the equation
3y23=03 y_{2} - 3 = 0
its solution
The graph
Rapid solution [src]
       3*(-1 + x2 + y2)
y3_1 = ----------------
              x2       
y31=3(x2+y21)x2y_{3 1} = \frac{3 \left(x_{2} + y_{2} - 1\right)}{x_{2}}
Sum and product of roots [src]
sum
3*(-1 + x2 + y2)
----------------
       x2       
(3(x2+y21)x2)\left(\frac{3 \left(x_{2} + y_{2} - 1\right)}{x_{2}}\right)
=
3*(-1 + x2 + y2)
----------------
       x2       
3(x2+y21)x2\frac{3 \left(x_{2} + y_{2} - 1\right)}{x_{2}}
product
3*(-1 + x2 + y2)
----------------
       x2       
(3(x2+y21)x2)\left(\frac{3 \left(x_{2} + y_{2} - 1\right)}{x_{2}}\right)
=
3*(-1 + x2 + y2)
----------------
       x2       
3(x2+y21)x2\frac{3 \left(x_{2} + y_{2} - 1\right)}{x_{2}}