Solve xy^2x-y=9 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = \frac{9 + y}{∣ y ∣} x = - \frac{9 + y}{∣ y ∣}

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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{2 x y ^{2}}{2 x ^{2} y - 1}

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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{16 x ^{4} y ^{4} - 8 x ^{2} y ^{3} - 2 y ^{2}}{8 x ^{6} y ^{3} - 12 x ^{4} y ^{2} + 6 x ^{2} y - 1}

Calculate

x y^{2} x - y = 9

Simplify the expression

x^{2} y^{2} - y = 9

Take the derivative of both sides

\frac{d}{d x} (x^{2} y^{2} - y) = \frac{d}{d x} (9)

Calculate the derivative

More Steps

2 x y^{2} + 2 x^{2} y \frac{d y}{d x} - \frac{d y}{d x} = \frac{d}{d x} (9)

Calculate the derivative

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

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

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

Move the constant to the right side

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

Removing 0 doesn't change the value,so remove it from the expression

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

Divide both sides

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

Divide the numbers

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

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

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

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{16 x ^{4} y ^{4} - 8 x ^{2} y ^{3} - 2 y ^{2}}{8 x ^{6} y ^{3} - 12 x ^{4} y ^{2} + 6 x ^{2} y - 1}

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