Solve xy^2y^(2)-2x-5y-1=0

Question

Solve the equation

x = \frac{5 y + 1}{y ^{4} - 2}

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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

Not 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{- y ^{4} + 2}{4 x y ^{3} - 5}

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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{20 y ^{10} x - 16 y ^{6} x - 40 y ^{7} + 80 y ^{3} - 48 x y ^{2}}{64 x ^{3} y ^{9} - 240 x ^{2} y ^{6} + 300 x y ^{3} - 125}

Calculate

x y^{2} y^{2} - 2 x - 5 y - 1 = 0

Simplify the expression

x y^{4} - 2 x - 5 y - 1 = 0

Take the derivative of both sides

\frac{d}{d x} (x y^{4} - 2 x - 5 y - 1) = \frac{d}{d x} (0)

Calculate the derivative

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y^{4} + 4 x y^{3} \frac{d y}{d x} - 2 - 5 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

y^{4} + 4 x y^{3} \frac{d y}{d x} - 2 - 5 \frac{d y}{d x} = 0

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

y^{4} - 2 + (4 x y^{3} - 5) \frac{d y}{d x} = 0

Move the constant to the right side

(4 x y^{3} - 5) \frac{d y}{d x} = 0 - (y^{4} - 2)

Subtract the terms

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(4 x y^{3} - 5) \frac{d y}{d x} = - y^{4} + 2

Divide both sides

\frac{( 4 x y ^{3} - 5 ) \frac{d y}{d x}}{4 x y ^{3} - 5} = \frac{- y ^{4} + 2}{4 x y ^{3} - 5}

Divide the numbers

\frac{d y}{d x} = \frac{- y ^{4} + 2}{4 x y ^{3} - 5}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{- y ^{4} + 2}{4 x y ^{3} - 5})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{- y ^{4} + 2}{4 x y ^{3} - 5})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( - y ^{4} + 2 ) \times ( 4 x y ^{3} - 5 ) - ( - y ^{4} + 2 ) \times \frac{d}{d x} ( 4 x y ^{3} - 5 )}{( 4 x y ^{3} - 5 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{3} \frac{d y}{d x} \times ( 4 x y ^{3} - 5 ) - ( - y ^{4} + 2 ) \times \frac{d}{d x} ( 4 x y ^{3} - 5 )}{( 4 x y ^{3} - 5 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{3} \frac{d y}{d x} \times ( 4 x y ^{3} - 5 ) - ( - y ^{4} + 2 ) ( 4 y ^{3} + 12 x y ^{2} \frac{d y}{d x} )}{( 4 x y ^{3} - 5 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 16 y ^{6} x \frac{d y}{d x} + 20 y ^{3} \frac{d y}{d x} - ( - y ^{4} + 2 ) ( 4 y ^{3} + 12 x y ^{2} \frac{d y}{d x} )}{( 4 x y ^{3} - 5 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 16 y ^{6} x \frac{d y}{d x} + 20 y ^{3} \frac{d y}{d x} - ( - 4 y ^{7} - 12 y ^{6} x \frac{d y}{d x} + 8 y ^{3} + 24 x y ^{2} \frac{d y}{d x} )}{( 4 x y ^{3} - 5 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{6} x \frac{d y}{d x} + 20 y ^{3} \frac{d y}{d x} + 4 y ^{7} - 8 y ^{3} - 24 x y ^{2} \frac{d y}{d x}}{( 4 x y ^{3} - 5 ) ^{2}}

Use equation \frac{d y}{d x} = \frac{- y ^{4} + 2}{4 x y ^{3} - 5} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{6} x \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5} + 20 y ^{3} \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5} + 4 y ^{7} - 8 y ^{3} - 24 x y ^{2} \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5}}{( 4 x y ^{3} - 5 ) ^{2}}

Solution

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Calculate

\frac{- 4 y ^{6} x \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5} + 20 y ^{3} \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5} + 4 y ^{7} - 8 y ^{3} - 24 x y ^{2} \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5}}{( 4 x y ^{3} - 5 ) ^{2}}

Multiply the terms

\frac{- \frac{4 y ^{6} x ( - y ^{4} + 2 )}{4 x y ^{3} - 5} + 20 y ^{3} \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5} + 4 y ^{7} - 8 y ^{3} - 24 x y ^{2} \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5}}{( 4 x y ^{3} - 5 ) ^{2}}

Multiply the terms

\frac{- \frac{4 y ^{6} x ( - y ^{4} + 2 )}{4 x y ^{3} - 5} + \frac{20 y ^{3} ( - y ^{4} + 2 )}{4 x y ^{3} - 5} + 4 y ^{7} - 8 y ^{3} - 24 x y ^{2} \times \frac{- y ^{4} + 2}{4 x y ^{3} - 5}}{( 4 x y ^{3} - 5 ) ^{2}}

Multiply the terms

\frac{- \frac{4 y ^{6} x ( - y ^{4} + 2 )}{4 x y ^{3} - 5} + \frac{20 y ^{3} ( - y ^{4} + 2 )}{4 x y ^{3} - 5} + 4 y ^{7} - 8 y ^{3} - \frac{24 x y ^{2} ( - y ^{4} + 2 )}{4 x y ^{3} - 5}}{( 4 x y ^{3} - 5 ) ^{2}}

Calculate the sum or difference

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\frac{\frac{20 y ^{10} x - 16 y ^{6} x - 40 y ^{7} + 80 y ^{3} - 48 x y ^{2}}{4 x y ^{3} - 5}}{( 4 x y ^{3} - 5 ) ^{2}}

Multiply by the reciprocal

\frac{20 y ^{10} x - 16 y ^{6} x - 40 y ^{7} + 80 y ^{3} - 48 x y ^{2}}{4 x y ^{3} - 5} \times \frac{1}{( 4 x y ^{3} - 5 ) ^{2}}

Multiply the terms

\frac{20 y ^{10} x - 16 y ^{6} x - 40 y ^{7} + 80 y ^{3} - 48 x y ^{2}}{( 4 x y ^{3} - 5 ) ( 4 x y ^{3} - 5 ) ^{2}}

Multiply the terms

\frac{20 y ^{10} x - 16 y ^{6} x - 40 y ^{7} + 80 y ^{3} - 48 x y ^{2}}{( 4 x y ^{3} - 5 ) ^{3}}

Expand the expression

More Steps

\frac{20 y ^{10} x - 16 y ^{6} x - 40 y ^{7} + 80 y ^{3} - 48 x y ^{2}}{64 x ^{3} y ^{9} - 240 x ^{2} y ^{6} + 300 x y ^{3} - 125}

\frac{d ^{2} y}{d x ^{2}} = \frac{20 y ^{10} x - 16 y ^{6} x - 40 y ^{7} + 80 y ^{3} - 48 x y ^{2}}{64 x ^{3} y ^{9} - 240 x ^{2} y ^{6} + 300 x y ^{3} - 125}

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