Solve (x*y) (y-1)=30

Question

Solve the equation

Solve for x

Solve for y

x = \frac{30}{y ^{2} - y}

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

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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{6 y ^{4} - 12 y ^{3} + 8 y ^{2} - 2 y}{8 y ^{3} x ^{2} - 12 y ^{2} x ^{2} + 6 y x ^{2} - x ^{2}}

Calculate

(x \cdot y) (y - 1) = 30

Simplify the expression

x y (y - 1) = 30

Take the derivative of both sides

\frac{d}{d x} (x y (y - 1)) = \frac{d}{d x} (30)

Calculate the derivative

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y^{2} - y + x y \frac{d y}{d x} - x \frac{d y}{d x} + x y \frac{d y}{d x} = \frac{d}{d x} (30)

Calculate the derivative

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

Calculate the sum or difference

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

Move the constant to the right side

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

Subtract the terms

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

Divide both sides

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

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

Calculate

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

Calculate

More Steps

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

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

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

Solution

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Calculate

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

Multiply the terms

\frac{- \frac{2 y ^{2} ( - y ^{2} + y )}{2 y - 1} + 2 y x \times \frac{- y ^{2} + y}{2 x y - x} - x \times \frac{- y ^{2} + y}{2 x y - x} + 2 y ^{3} - 3 y ^{2} + y}{( 2 x y - x ) ^{2}}

Multiply the terms

More Steps

\frac{- \frac{2 y ^{2} ( - y ^{2} + y )}{2 y - 1} + \frac{2 y ( - y ^{2} + y )}{2 y - 1} - x \times \frac{- y ^{2} + y}{2 x y - x} + 2 y ^{3} - 3 y ^{2} + y}{( 2 x y - x ) ^{2}}

Multiply the terms

More Steps

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

Calculate the sum or difference

More Steps

\frac{\frac{6 y ^{4} - 12 y ^{3} + 8 y ^{2} - 2 y}{2 y - 1}}{( 2 x y - x ) ^{2}}

Multiply by the reciprocal

\frac{6 y ^{4} - 12 y ^{3} + 8 y ^{2} - 2 y}{2 y - 1} \times \frac{1}{( 2 x y - x ) ^{2}}

Multiply the terms

\frac{6 y ^{4} - 12 y ^{3} + 8 y ^{2} - 2 y}{( 2 y - 1 ) ( 2 x y - x ) ^{2}}

Expand the expression

More Steps

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

\frac{d ^{2} y}{d x ^{2}} = \frac{6 y ^{4} - 12 y ^{3} + 8 y ^{2} - 2 y}{8 y ^{3} x ^{2} - 12 y ^{2} x ^{2} + 6 y x ^{2} - x ^{2}}

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