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

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} = 0

Multiply the terms

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

Rewrite the expression

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

Calculate

More Steps

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

Show Solution

Testing for symmetry

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

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x}

Calculate

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

Simplify the expression

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

Take the derivative of both sides

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

Calculate the derivative

More Steps

6 x y^{2} + 6 x^{2} y \frac{d y}{d x} - 2 x y^{3} - 3 x^{2} y^{2} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

6 x y^{2} + 6 x^{2} y \frac{d y}{d x} - 2 x y^{3} - 3 x^{2} y^{2} \frac{d y}{d x} = 0

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

6 x y^{2} - 2 x y^{3} + (6 x^{2} y - 3 x^{2} y^{2}) \frac{d y}{d x} = 0

Move the constant to the right side

(6 x^{2} y - 3 x^{2} y^{2}) \frac{d y}{d x} = 0 - (6 x y^{2} - 2 x y^{3})

Subtract the terms

More Steps

(6 x^{2} y - 3 x^{2} y^{2}) \frac{d y}{d x} = - 6 x y^{2} + 2 x y^{3}

Divide both sides

\frac{( 6 x ^{2} y - 3 x ^{2} y ^{2} ) \frac{d y}{d x}}{6 x ^{2} y - 3 x ^{2} y ^{2}} = \frac{- 6 x y ^{2} + 2 x y ^{3}}{6 x ^{2} y - 3 x ^{2} y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{- 6 x y ^{2} + 2 x y ^{3}}{6 x ^{2} y - 3 x ^{2} y ^{2}}

Solution

More Steps

\frac{d y}{d x} = \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x}

Show Solution

Find the second derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{- 144 y + 168 y ^{2} - 70 y ^{3} + 10 y ^{4}}{108 y x ^{2} - 54 y ^{2} x ^{2} + 9 y ^{3} x ^{2} - 72 x ^{2}}

Calculate

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

Simplify the expression

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

Take the derivative of both sides

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

Calculate the derivative

More Steps

6 x y^{2} + 6 x^{2} y \frac{d y}{d x} - 2 x y^{3} - 3 x^{2} y^{2} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

6 x y^{2} + 6 x^{2} y \frac{d y}{d x} - 2 x y^{3} - 3 x^{2} y^{2} \frac{d y}{d x} = 0

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

6 x y^{2} - 2 x y^{3} + (6 x^{2} y - 3 x^{2} y^{2}) \frac{d y}{d x} = 0

Move the constant to the right side

(6 x^{2} y - 3 x^{2} y^{2}) \frac{d y}{d x} = 0 - (6 x y^{2} - 2 x y^{3})

Subtract the terms

More Steps

(6 x^{2} y - 3 x^{2} y^{2}) \frac{d y}{d x} = - 6 x y^{2} + 2 x y^{3}

Divide both sides

\frac{( 6 x ^{2} y - 3 x ^{2} y ^{2} ) \frac{d y}{d x}}{6 x ^{2} y - 3 x ^{2} y ^{2}} = \frac{- 6 x y ^{2} + 2 x y ^{3}}{6 x ^{2} y - 3 x ^{2} y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{- 6 x y ^{2} + 2 x y ^{3}}{6 x ^{2} y - 3 x ^{2} y ^{2}}

Divide the numbers

More Steps

\frac{d y}{d x} = \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{- 6 y + 2 y ^{2}}{6 x - 3 y x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{- 6 y + 2 y ^{2}}{6 x - 3 y x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( - 6 y + 2 y ^{2} ) \times ( 6 x - 3 y x ) - ( - 6 y + 2 y ^{2} ) \times \frac{d}{d x} ( 6 x - 3 y x )}{( 6 x - 3 y x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{( - 6 \frac{d y}{d x} + 4 y \frac{d y}{d x} ) ( 6 x - 3 y x ) - ( - 6 y + 2 y ^{2} ) \times \frac{d}{d x} ( 6 x - 3 y x )}{( 6 x - 3 y x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{( - 6 \frac{d y}{d x} + 4 y \frac{d y}{d x} ) ( 6 x - 3 y x ) - ( - 6 y + 2 y ^{2} ) ( 6 - 3 y - 3 x \frac{d y}{d x} )}{( 6 x - 3 y x ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 36 x \frac{d y}{d x} + 42 y x \frac{d y}{d x} - 12 y ^{2} x \frac{d y}{d x} - ( - 6 y + 2 y ^{2} ) ( 6 - 3 y - 3 x \frac{d y}{d x} )}{( 6 x - 3 y x ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 36 x \frac{d y}{d x} + 42 y x \frac{d y}{d x} - 12 y ^{2} x \frac{d y}{d x} - ( - 36 y + 30 y ^{2} + 18 y x \frac{d y}{d x} - 6 y ^{3} - 6 y ^{2} x \frac{d y}{d x} )}{( 6 x - 3 y x ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 36 x \frac{d y}{d x} + 24 y x \frac{d y}{d x} - 6 y ^{2} x \frac{d y}{d x} + 36 y - 30 y ^{2} + 6 y ^{3}}{( 6 x - 3 y x ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{- 12 x \frac{d y}{d x} + 8 y x \frac{d y}{d x} - 2 y ^{2} x \frac{d y}{d x} + 12 y - 10 y ^{2} + 2 y ^{3}}{3 ( 2 x - y x ) ^{2}}

Use equation \frac{d y}{d x} = \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- 12 x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} + 8 y x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} - 2 y ^{2} x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} + 12 y - 10 y ^{2} + 2 y ^{3}}{3 ( 2 x - y x ) ^{2}}

Solution

More Steps

Calculate

\frac{- 12 x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} + 8 y x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} - 2 y ^{2} x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} + 12 y - 10 y ^{2} + 2 y ^{3}}{3 ( 2 x - y x ) ^{2}}

Multiply the terms

\frac{- \frac{4 ( - 6 y + 2 y ^{2} )}{2 - y} + 8 y x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} - 2 y ^{2} x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} + 12 y - 10 y ^{2} + 2 y ^{3}}{3 ( 2 x - y x ) ^{2}}

Multiply the terms

More Steps

\frac{- \frac{4 ( - 6 y + 2 y ^{2} )}{2 - y} + \frac{8 y ( - 6 y + 2 y ^{2} )}{6 - 3 y} - 2 y ^{2} x \times \frac{- 6 y + 2 y ^{2}}{6 x - 3 y x} + 12 y - 10 y ^{2} + 2 y ^{3}}{3 ( 2 x - y x ) ^{2}}

Multiply the terms

\frac{- \frac{4 ( - 6 y + 2 y ^{2} )}{2 - y} + \frac{8 y ( - 6 y + 2 y ^{2} )}{6 - 3 y} - \frac{2 y ^{2} ( - 6 y + 2 y ^{2} )}{6 - 3 y} + 12 y - 10 y ^{2} + 2 y ^{3}}{3 ( 2 x - y x ) ^{2}}

Calculate the sum or difference

More Steps

\frac{\frac{- 144 y + 168 y ^{2} - 70 y ^{3} + 10 y ^{4}}{3 ( y - 2 )}}{3 ( 2 x - y x ) ^{2}}

Multiply by the reciprocal

\frac{- 144 y + 168 y ^{2} - 70 y ^{3} + 10 y ^{4}}{3 ( y - 2 )} \times \frac{1}{3 ( 2 x - y x ) ^{2}}

Multiply the terms

\frac{- 144 y + 168 y ^{2} - 70 y ^{3} + 10 y ^{4}}{3 ( y - 2 ) \times 3 ( 2 x - y x ) ^{2}}

Multiply the terms

\frac{- 144 y + 168 y ^{2} - 70 y ^{3} + 10 y ^{4}}{9 ( y - 2 ) ( 2 x - y x ) ^{2}}

Expand the expression

More Steps

\frac{- 144 y + 168 y ^{2} - 70 y ^{3} + 10 y ^{4}}{108 y x ^{2} - 54 y ^{2} x ^{2} + 9 y ^{3} x ^{2} - 72 x ^{2}}

\frac{d ^{2} y}{d x ^{2}} = \frac{- 144 y + 168 y ^{2} - 70 y ^{3} + 10 y ^{4}}{108 y x ^{2} - 54 y ^{2} x ^{2} + 9 y ^{3} x ^{2} - 72 x ^{2}}

Show Solution