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

Question

Solve the equation

Solve for x

Solve for y

x = - y^{2} + 1 x = - - y^{2} + 1

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Testing for symmetry

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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Rewrite the equation

r = 0 r = 1 r = - 1

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3}

Calculate

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

Simplify the expression

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

Take the derivative of both sides

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

Calculate the derivative

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4 x y^{3} + 4 y^{4} \frac{d y}{d x} + 6 x^{2} y^{2} \frac{d y}{d x} + 6 y^{4} \frac{d y}{d x} - 6 y^{2} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

4 x y^{3} + 4 y^{4} \frac{d y}{d x} + 6 x^{2} y^{2} \frac{d y}{d x} + 6 y^{4} \frac{d y}{d x} - 6 y^{2} \frac{d y}{d x} = 0

Calculate the sum or difference

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4 x y^{3} + (10 y^{4} + 6 x^{2} y^{2} - 6 y^{2}) \frac{d y}{d x} = 0

Move the constant to the right side

(10 y^{4} + 6 x^{2} y^{2} - 6 y^{2}) \frac{d y}{d x} = 0 - 4 x y^{3}

Removing 0 doesn't change the value,so remove it from the expression

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

Divide both sides

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

Divide the numbers

\frac{d y}{d x} = \frac{- 4 x y ^{3}}{10 y ^{4} + 6 x ^{2} y ^{2} - 6 y ^{2}}

Solution

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\frac{d y}{d x} = - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = - \frac{50 y ^{5} + 20 x ^{2} y ^{3} - 60 y ^{3} - 30 x ^{4} y + 12 x ^{2} y + 18 y}{125 y ^{6} + 27 x ^{6} - 27 + 225 y ^{4} x ^{2} - 225 y ^{4} + 135 x ^{4} y ^{2} - 81 x ^{4} + 135 y ^{2} + 81 x ^{2} - 270 y ^{2} x ^{2}}

Calculate

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

Simplify the expression

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

Take the derivative of both sides

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

Calculate the derivative

More Steps

4 x y^{3} + 4 y^{4} \frac{d y}{d x} + 6 x^{2} y^{2} \frac{d y}{d x} + 6 y^{4} \frac{d y}{d x} - 6 y^{2} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

4 x y^{3} + 4 y^{4} \frac{d y}{d x} + 6 x^{2} y^{2} \frac{d y}{d x} + 6 y^{4} \frac{d y}{d x} - 6 y^{2} \frac{d y}{d x} = 0

Calculate the sum or difference

More Steps

4 x y^{3} + (10 y^{4} + 6 x^{2} y^{2} - 6 y^{2}) \frac{d y}{d x} = 0

Move the constant to the right side

(10 y^{4} + 6 x^{2} y^{2} - 6 y^{2}) \frac{d y}{d x} = 0 - 4 x y^{3}

Removing 0 doesn't change the value,so remove it from the expression

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

Divide both sides

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

Divide the numbers

\frac{d y}{d x} = \frac{- 4 x y ^{3}}{10 y ^{4} + 6 x ^{2} y ^{2} - 6 y ^{2}}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 2 x y ) \times ( 5 y ^{2} + 3 x ^{2} - 3 ) - 2 x y \times \frac{d}{d x} ( 5 y ^{2} + 3 x ^{2} - 3 )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{( 2 y + 2 x \frac{d y}{d x} ) ( 5 y ^{2} + 3 x ^{2} - 3 ) - 2 x y \times \frac{d}{d x} ( 5 y ^{2} + 3 x ^{2} - 3 )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{( 2 y + 2 x \frac{d y}{d x} ) ( 5 y ^{2} + 3 x ^{2} - 3 ) - 2 x y ( 10 y \frac{d y}{d x} + 6 x )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{10 y ^{3} + 6 y x ^{2} - 6 y + 10 x y ^{2} \frac{d y}{d x} + 6 x ^{3} \frac{d y}{d x} - 6 x \frac{d y}{d x} - 2 x y ( 10 y \frac{d y}{d x} + 6 x )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{10 y ^{3} + 6 y x ^{2} - 6 y + 10 x y ^{2} \frac{d y}{d x} + 6 x ^{3} \frac{d y}{d x} - 6 x \frac{d y}{d x} - ( 20 x y ^{2} \frac{d y}{d x} + 12 x ^{2} y )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{10 y ^{3} - 6 y x ^{2} - 6 y - 10 x y ^{2} \frac{d y}{d x} + 6 x ^{3} \frac{d y}{d x} - 6 x \frac{d y}{d x}}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{10 y ^{3} - 6 y x ^{2} - 6 y - 10 x y ^{2} ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} ) + 6 x ^{3} ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} ) - 6 x ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Solution

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Calculate

- \frac{10 y ^{3} - 6 y x ^{2} - 6 y - 10 x y ^{2} ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} ) + 6 x ^{3} ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} ) - 6 x ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Multiply

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- \frac{10 y ^{3} - 6 y x ^{2} - 6 y + \frac{20 x ^{2} y ^{3}}{5 y ^{2} + 3 x ^{2} - 3} + 6 x ^{3} ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} ) - 6 x ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Multiply

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- \frac{10 y ^{3} - 6 y x ^{2} - 6 y + \frac{20 x ^{2} y ^{3}}{5 y ^{2} + 3 x ^{2} - 3} - \frac{12 x ^{4} y}{5 y ^{2} + 3 x ^{2} - 3} - 6 x ( - \frac{2 x y}{5 y ^{2} + 3 x ^{2} - 3} )}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Multiply

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- \frac{10 y ^{3} - 6 y x ^{2} - 6 y + \frac{20 x ^{2} y ^{3}}{5 y ^{2} + 3 x ^{2} - 3} - \frac{12 x ^{4} y}{5 y ^{2} + 3 x ^{2} - 3} + \frac{12 x ^{2} y}{5 y ^{2} + 3 x ^{2} - 3}}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Calculate the sum or difference

More Steps

- \frac{\frac{50 y ^{5} + 20 x ^{2} y ^{3} - 60 y ^{3} - 30 x ^{4} y + 12 x ^{2} y + 18 y}{5 y ^{2} + 3 x ^{2} - 3}}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{2}}

Divide the terms

More Steps

- \frac{50 y ^{5} + 20 x ^{2} y ^{3} - 60 y ^{3} - 30 x ^{4} y + 12 x ^{2} y + 18 y}{( 5 y ^{2} + 3 x ^{2} - 3 ) ^{3}}

Evaluate the power

More Steps

- \frac{50 y ^{5} + 20 x ^{2} y ^{3} - 60 y ^{3} - 30 x ^{4} y + 12 x ^{2} y + 18 y}{125 y ^{6} + 27 x ^{6} - 27 + 225 y ^{4} x ^{2} - 225 y ^{4} + 135 x ^{4} y ^{2} - 81 x ^{4} + 135 y ^{2} + 81 x ^{2} - 270 y ^{2} x ^{2}}

\frac{d ^{2} y}{d x ^{2}} = - \frac{50 y ^{5} + 20 x ^{2} y ^{3} - 60 y ^{3} - 30 x ^{4} y + 12 x ^{2} y + 18 y}{125 y ^{6} + 27 x ^{6} - 27 + 225 y ^{4} x ^{2} - 225 y ^{4} + 135 x ^{4} y ^{2} - 81 x ^{4} + 135 y ^{2} + 81 x ^{2} - 270 y ^{2} x ^{2}}

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