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

Question

Solve the equation

x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y > 3 x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y > 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0

Evaluate

(x^{2} y^{2} - 1) \times 3 = x^{2} y^{3}

Multiply the terms

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

Rewrite the expression

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

Expand the expression

More Steps

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

Move the expression to the left side

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

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

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

Move the constant to the right side

(3 y^{2} - y^{3}) x^{2} = 3

Divide both sides

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

Divide the numbers

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

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm \frac{3}{3 y ^{2} - y ^{3}}

Simplify the expression

More Steps

x = \pm \frac{9 - 3 y}{∣ y ∣ ∣ 3 - y ∣}

Separate the equation into 2 possible cases

x = \frac{9 - 3 y}{∣ y ∣ ∣ 3 - y ∣} x = - \frac{9 - 3 y}{∣ y ∣ ∣ 3 - y ∣}

Multiply the terms

x = \frac{9 - 3 y}{∣ y ( 3 - y ) ∣} x = - \frac{9 - 3 y}{∣ y ∣ ∣ 3 - y ∣}

Multiply the terms

x = \frac{9 - 3 y}{∣ y ( 3 - y ) ∣} x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}

Calculate

{x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣} 0 < y < 3 {x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣} y > 3 {x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣} y < 0 {x = \frac{9 - 3 y}{∣ y ( 3 - y ) ∣} 0 < y < 3 {x = \frac{9 - 3 y}{∣ y ( 3 - y ) ∣} y > 3 {x = \frac{9 - 3 y}{∣ y ( 3 - y ) ∣} y < 0

Simplify

x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, 0 < y < 3 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y > 3 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y < 0 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0 x = \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, 0 < y < 3 x = \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y > 3

Simplify

x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, 0 < y < 3 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y > 3 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y < 0 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0 x = \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y > 3

Simplify

x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, 0 < y < 3 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y > 3 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y < 0 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y > 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0

Simplify

x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, 0 < y < 3 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y > 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y > 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0

Simplify

x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0 x = - \frac{9 - 3 y}{∣ y ( 3 - y ) ∣}, y > 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y > 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0

Solution

x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y > 3 x = - \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, 0 < y < 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y > 3 x = \frac{9 - 3 y}{∣ 3 y - y ^{2} ∣}, y < 0

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not 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{2 y ^{2} - 6 y}{6 x - 3 y x}

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