Solve xy y-x=7 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = \frac{7}{y ^{2} - 1}

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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 derivative with respect to x

Find the derivative with respect to y

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

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Find the second derivative with respect to x

Find the second derivative with respect to y

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

Calculate

x yy - x = 7

Simplify the expression

x y^{2} - x = 7

Take the derivative of both sides

\frac{d}{d x} (x y^{2} - x) = \frac{d}{d x} (7)

Calculate the derivative

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y^{2} + 2 x y \frac{d y}{d x} - 1 = \frac{d}{d x} (7)

Calculate the derivative

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

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

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

Subtract the terms

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

Divide both sides

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

Calculate

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

Use equation \frac{d y}{d x} = \frac{- y ^{2} + 1}{2 x y} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- y ^{2} x \times \frac{- y ^{2} + 1}{2 x y} + y ^{3} - y - x \times \frac{- y ^{2} + 1}{2 x y}}{2 x ^{2} y ^{2}}

Solution

More Steps

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

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