Solve x^2-4xy y^2=4

Question

Solve the equation

Solve for x

Solve for y

x = 2 y^{3} + 2 y^{6} + 1 x = 2 y^{3} - 2 y^{6} + 1

Evaluate

x^{2} - 4 x y \times y^{2} = 4

Multiply

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x^{2} - 4 x y^{3} = 4

Rewrite the expression

x^{2} - 4 y^{3} x = 4

Move the expression to the left side

x^{2} - 4 y^{3} x - 4 = 0

Substitute a= 1,b= - 4 y^{3} and c= - 4 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

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

Simplify the expression

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x = \frac{4 y ^{3} \pm 16 y ^{6} + 16}{2}

Simplify the radical expression

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x = \frac{4 y ^{3} \pm 4 y ^{6} + 1}{2}

Separate the equation into 2 possible cases

x = \frac{4 y ^{3} + 4 y ^{6} + 1}{2} x = \frac{4 y ^{3} - 4 y ^{6} + 1}{2}

Simplify the expression

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x = 2 y^{3} + 2 y^{6} + 1 x = \frac{4 y ^{3} - 4 y ^{6} + 1}{2}

Solution

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x = 2 y^{3} + 2 y^{6} + 1 x = 2 y^{3} - 2 y^{6} + 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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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

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

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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{y ^{3} x + 8 y ^{6} - x ^{2}}{18 y ^{5} x ^{2}}

Calculate

x^{2} - 4 x y y^{2} = 4

Simplify the expression

x^{2} - 4 x y^{3} = 4

Take the derivative of both sides

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

Calculate the derivative

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

Calculate the derivative

2 x - 4 y^{3} - 12 x y^{2} \frac{d y}{d x} = 0

Move the expression to the right-hand side and change its sign

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

Subtract the terms

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 1 - 6 y ^{2} \frac{d y}{d x} ) \times 6 x y ^{2} - ( x - 2 y ^{3} ) \times \frac{d}{d x} ( 6 x y ^{2} )}{( 6 x y ^{2} ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 1 - 6 y ^{2} \frac{d y}{d x} ) \times 6 x y ^{2} - ( x - 2 y ^{3} ) ( 6 y ^{2} + 12 x y \frac{d y}{d x} )}{( 6 x y ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{6 x y ^{2} - 36 y ^{4} x \frac{d y}{d x} - ( x - 2 y ^{3} ) ( 6 y ^{2} + 12 x y \frac{d y}{d x} )}{( 6 x y ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{6 x y ^{2} - 36 y ^{4} x \frac{d y}{d x} - ( 6 x y ^{2} - 12 y ^{5} + 12 x ^{2} y \frac{d y}{d x} - 24 y ^{4} x \frac{d y}{d x} )}{( 6 x y ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 12 y ^{4} x \frac{d y}{d x} + 12 y ^{5} - 12 x ^{2} y \frac{d y}{d x}}{( 6 x y ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 12 y ^{4} x \frac{d y}{d x} + 12 y ^{5} - 12 x ^{2} y \frac{d y}{d x}}{36 x ^{2} y ^{4}}

Calculate

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

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

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

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{y ^{3} x + 8 y ^{6} - x ^{2}}{18 y ^{5} x ^{2}}

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