Solve x y x - y = 6012

Question

Solve the equation

Solve for x

Solve for y

x = \frac{6012 y + y ^{2}}{∣ y ∣} x = - \frac{6012 y + y ^{2}}{∣ 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}{x ^{2} - 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{6 x ^{2} y + 2 y}{x ^{4} - 2 x ^{2} + 1}

Calculate

x y x - y = 6012

Simplify the expression

x^{2} y - y = 6012

Take the derivative of both sides

\frac{d}{d x} (x^{2} y - y) = \frac{d}{d x} (6012)

Calculate the derivative

More Steps

2 x y + x^{2} \frac{d y}{d x} - \frac{d y}{d x} = \frac{d}{d x} (6012)

Calculate the derivative

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

Move the constant to the right side

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

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

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

Divide both sides

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

Divide the numbers

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

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

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

Calculate

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

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

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

Solution

More Steps

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

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