Solve (x^2 + y^2 -1)3 - x^2y^2 = 0

Question

Solve the equation

Solve for x

Solve for y

x = \frac{- 12 y ^{2} + 3 y ^{4} + 9}{∣ 3 - y ^{2} ∣} x = - \frac{- 12 y ^{2} + 3 y ^{4} + 9}{∣ 3 - y ^{2} ∣}

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{- 3 x + x y ^{2}}{3 y - x ^{2} y}

Calculate

(x^{2} + y^{2} - 1) 3 - x^{2} y^{2} = 0

Simplify the expression

3 x^{2} + 3 y^{2} - 3 - x^{2} y^{2} = 0

Take the derivative of both sides

\frac{d}{d x} (3 x^{2} + 3 y^{2} - 3 - x^{2} y^{2}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

6 x + 6 y \frac{d y}{d x} - 2 x y^{2} - 2 x^{2} y \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

6 x + 6 y \frac{d y}{d x} - 2 x y^{2} - 2 x^{2} y \frac{d y}{d x} = 0

Collect like terms by calculating the sum or difference of their coefficients

6 x - 2 x y^{2} + (6 y - 2 x^{2} y) \frac{d y}{d x} = 0

Move the constant to the right side

(6 y - 2 x^{2} y) \frac{d y}{d x} = 0 - (6 x - 2 x y^{2})

Subtract the terms

More Steps

(6 y - 2 x^{2} y) \frac{d y}{d x} = - 6 x + 2 x y^{2}

Divide both sides

\frac{( 6 y - 2 x ^{2} y ) \frac{d y}{d x}}{6 y - 2 x ^{2} y} = \frac{- 6 x + 2 x y ^{2}}{6 y - 2 x ^{2} y}

Divide the numbers

\frac{d y}{d x} = \frac{- 6 x + 2 x y ^{2}}{6 y - 2 x ^{2} y}

Solution

More Steps

\frac{d y}{d x} = \frac{- 3 x + x y ^{2}}{3 y - x ^{2} y}

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