Solve x^4 - y^4 = xy

Question

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 = \frac{2 tan ( 2 θ )}{2} r = - \frac{2 tan ( 2 θ )}{2}

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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{- y + 4 x ^{3}}{4 y ^{3} + x}

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Find the second derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{4} + 160 x ^{3} y ^{3} + 2 x y + 4 x ^{4} + 192 y ^{6} x ^{2} - 192 x ^{6} y ^{2}}{64 y ^{9} + 48 y ^{6} x + 12 y ^{3} x ^{2} + x ^{3}}

Calculate

x^{4} - y^{4} = x y

Take the derivative of both sides

\frac{d}{d x} (x^{4} - y^{4}) = \frac{d}{d x} (x y)

Calculate the derivative

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

Calculate the derivative

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

Move the expression to the left side

4 x^{3} - 4 y^{3} \frac{d y}{d x} - x \frac{d y}{d x} = y

Move the expression to the right side

- 4 y^{3} \frac{d y}{d x} - x \frac{d y}{d x} = y - 4 x^{3}

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

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

Divide both sides

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

Divide the numbers

\frac{d y}{d x} = \frac{y - 4 x ^{3}}{- 4 y ^{3} - x}

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( - \frac{d y}{d x} + 12 x ^{2} ) ( 4 y ^{3} + x ) - ( - y + 4 x ^{3} ) \times \frac{d}{d x} ( 4 y ^{3} + x )}{( 4 y ^{3} + x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( - \frac{d y}{d x} + 12 x ^{2} ) ( 4 y ^{3} + x ) - ( - y + 4 x ^{3} ) ( 12 y ^{2} \frac{d y}{d x} + 1 )}{( 4 y ^{3} + x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{3} \frac{d y}{d x} - x \frac{d y}{d x} + 48 x ^{2} y ^{3} + 12 x ^{3} - ( - y + 4 x ^{3} ) ( 12 y ^{2} \frac{d y}{d x} + 1 )}{( 4 y ^{3} + x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{3} \frac{d y}{d x} - x \frac{d y}{d x} + 48 x ^{2} y ^{3} + 12 x ^{3} - ( - 12 y ^{3} \frac{d y}{d x} + 48 x ^{3} y ^{2} \frac{d y}{d x} - y + 4 x ^{3} )}{( 4 y ^{3} + x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{8 y ^{3} \frac{d y}{d x} - x \frac{d y}{d x} + 48 x ^{2} y ^{3} + 8 x ^{3} - 48 x ^{3} y ^{2} \frac{d y}{d x} + y}{( 4 y ^{3} + x ) ^{2}}

Use equation \frac{d y}{d x} = \frac{- y + 4 x ^{3}}{4 y ^{3} + x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{8 y ^{3} \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} - x \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} + 48 x ^{2} y ^{3} + 8 x ^{3} - 48 x ^{3} y ^{2} \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} + y}{( 4 y ^{3} + x ) ^{2}}

Solution

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Calculate

\frac{8 y ^{3} \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} - x \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} + 48 x ^{2} y ^{3} + 8 x ^{3} - 48 x ^{3} y ^{2} \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} + y}{( 4 y ^{3} + x ) ^{2}}

Multiply the terms

\frac{\frac{8 y ^{3} ( - y + 4 x ^{3} )}{4 y ^{3} + x} - x \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} + 48 x ^{2} y ^{3} + 8 x ^{3} - 48 x ^{3} y ^{2} \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} + y}{( 4 y ^{3} + x ) ^{2}}

Multiply the terms

\frac{\frac{8 y ^{3} ( - y + 4 x ^{3} )}{4 y ^{3} + x} - \frac{x ( - y + 4 x ^{3} )}{4 y ^{3} + x} + 48 x ^{2} y ^{3} + 8 x ^{3} - 48 x ^{3} y ^{2} \times \frac{- y + 4 x ^{3}}{4 y ^{3} + x} + y}{( 4 y ^{3} + x ) ^{2}}

Multiply the terms

\frac{\frac{8 y ^{3} ( - y + 4 x ^{3} )}{4 y ^{3} + x} - \frac{x ( - y + 4 x ^{3} )}{4 y ^{3} + x} + 48 x ^{2} y ^{3} + 8 x ^{3} - \frac{48 x ^{3} y ^{2} ( - y + 4 x ^{3} )}{4 y ^{3} + x} + y}{( 4 y ^{3} + x ) ^{2}}

Calculate the sum or difference

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\frac{\frac{- 4 y ^{4} + 160 x ^{3} y ^{3} + 2 x y + 4 x ^{4} + 192 y ^{6} x ^{2} - 192 x ^{6} y ^{2}}{4 y ^{3} + x}}{( 4 y ^{3} + x ) ^{2}}

Multiply by the reciprocal

\frac{- 4 y ^{4} + 160 x ^{3} y ^{3} + 2 x y + 4 x ^{4} + 192 y ^{6} x ^{2} - 192 x ^{6} y ^{2}}{4 y ^{3} + x} \times \frac{1}{( 4 y ^{3} + x ) ^{2}}

Multiply the terms

\frac{- 4 y ^{4} + 160 x ^{3} y ^{3} + 2 x y + 4 x ^{4} + 192 y ^{6} x ^{2} - 192 x ^{6} y ^{2}}{( 4 y ^{3} + x ) ( 4 y ^{3} + x ) ^{2}}

Multiply the terms

\frac{- 4 y ^{4} + 160 x ^{3} y ^{3} + 2 x y + 4 x ^{4} + 192 y ^{6} x ^{2} - 192 x ^{6} y ^{2}}{( 4 y ^{3} + x ) ^{3}}

Expand the expression

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\frac{- 4 y ^{4} + 160 x ^{3} y ^{3} + 2 x y + 4 x ^{4} + 192 y ^{6} x ^{2} - 192 x ^{6} y ^{2}}{64 y ^{9} + 48 y ^{6} x + 12 y ^{3} x ^{2} + x ^{3}}

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{4} + 160 x ^{3} y ^{3} + 2 x y + 4 x ^{4} + 192 y ^{6} x ^{2} - 192 x ^{6} y ^{2}}{64 y ^{9} + 48 y ^{6} x + 12 y ^{3} x ^{2} + x ^{3}}

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