Solve 3xy^2x-5y=6

Question

Solve the equation

Solve for x

Solve for y

x = - \frac{18 + 15 y}{3 ∣ y ∣}, y \neq = 0 x = \frac{18 + 15 y}{3 ∣ y ∣}, y \neq = 0

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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{6 x y ^{2}}{6 x ^{2} y - 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{432 x ^{4} y ^{4} - 360 x ^{2} y ^{3} - 150 y ^{2}}{216 x ^{6} y ^{3} - 540 x ^{4} y ^{2} + 450 x ^{2} y - 125}

Calculate

3 x y^{2} x - 5 y = 6

Simplify the expression

3 x^{2} y^{2} - 5 y = 6

Take the derivative of both sides

\frac{d}{d x} (3 x^{2} y^{2} - 5 y) = \frac{d}{d x} (6)

Calculate the derivative

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6 x y^{2} + 6 x^{2} y \frac{d y}{d x} - 5 \frac{d y}{d x} = \frac{d}{d x} (6)

Calculate the derivative

6 x y^{2} + 6 x^{2} y \frac{d y}{d x} - 5 \frac{d y}{d x} = 0

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

6 x y^{2} + (6 x^{2} y - 5) \frac{d y}{d x} = 0

Move the constant to the right side

(6 x^{2} y - 5) \frac{d y}{d x} = 0 - 6 x y^{2}

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

(6 x^{2} y - 5) \frac{d y}{d x} = - 6 x y^{2}

Divide both sides

\frac{( 6 x ^{2} y - 5 ) \frac{d y}{d x}}{6 x ^{2} y - 5} = \frac{- 6 x y ^{2}}{6 x ^{2} y - 5}

Divide the numbers

\frac{d y}{d x} = \frac{- 6 x y ^{2}}{6 x ^{2} y - 5}

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

\frac{d y}{d x} = - \frac{6 x y ^{2}}{6 x ^{2} y - 5}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{6 x y ^{2}}{6 x ^{2} y - 5})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{6 x y ^{2}}{6 x ^{2} y - 5})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 6 x y ^{2} ) \times ( 6 x ^{2} y - 5 ) - 6 x y ^{2} \times \frac{d}{d x} ( 6 x ^{2} y - 5 )}{( 6 x ^{2} y - 5 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{( 6 y ^{2} + 12 x y \frac{d y}{d x} ) ( 6 x ^{2} y - 5 ) - 6 x y ^{2} \times \frac{d}{d x} ( 6 x ^{2} y - 5 )}{( 6 x ^{2} y - 5 ) ^{2}}

Calculate the derivative

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

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{36 y ^{3} x ^{2} - 30 y ^{2} + 72 x ^{3} y ^{2} \frac{d y}{d x} - 60 x y \frac{d y}{d x} - 6 x y ^{2} ( 12 x y + 6 x ^{2} \frac{d y}{d x} )}{( 6 x ^{2} y - 5 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{36 y ^{3} x ^{2} - 30 y ^{2} + 72 x ^{3} y ^{2} \frac{d y}{d x} - 60 x y \frac{d y}{d x} - ( 72 x ^{2} y ^{3} + 36 x ^{3} y ^{2} \frac{d y}{d x} )}{( 6 x ^{2} y - 5 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{- 36 y ^{3} x ^{2} - 30 y ^{2} + 36 x ^{3} y ^{2} \frac{d y}{d x} - 60 x y \frac{d y}{d x}}{( 6 x ^{2} y - 5 ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{6 x y ^{2}}{6 x ^{2} y - 5} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 36 y ^{3} x ^{2} - 30 y ^{2} + 36 x ^{3} y ^{2} ( - \frac{6 x y ^{2}}{6 x ^{2} y - 5} ) - 60 x y ( - \frac{6 x y ^{2}}{6 x ^{2} y - 5} )}{( 6 x ^{2} y - 5 ) ^{2}}

Solution

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Calculate

- \frac{- 36 y ^{3} x ^{2} - 30 y ^{2} + 36 x ^{3} y ^{2} ( - \frac{6 x y ^{2}}{6 x ^{2} y - 5} ) - 60 x y ( - \frac{6 x y ^{2}}{6 x ^{2} y - 5} )}{( 6 x ^{2} y - 5 ) ^{2}}

Multiply

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- \frac{- 36 y ^{3} x ^{2} - 30 y ^{2} - \frac{216 x ^{4} y ^{4}}{6 x ^{2} y - 5} - 60 x y ( - \frac{6 x y ^{2}}{6 x ^{2} y - 5} )}{( 6 x ^{2} y - 5 ) ^{2}}

Multiply

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- \frac{- 36 y ^{3} x ^{2} - 30 y ^{2} - \frac{216 x ^{4} y ^{4}}{6 x ^{2} y - 5} + \frac{360 x ^{2} y ^{3}}{6 x ^{2} y - 5}}{( 6 x ^{2} y - 5 ) ^{2}}

Calculate the sum or difference

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- \frac{\frac{- 432 x ^{4} y ^{4} + 360 x ^{2} y ^{3} + 150 y ^{2}}{6 x ^{2} y - 5}}{( 6 x ^{2} y - 5 ) ^{2}}

Divide the terms

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- \frac{- 432 x ^{4} y ^{4} + 360 x ^{2} y ^{3} + 150 y ^{2}}{( 6 x ^{2} y - 5 ) ^{3}}

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

\frac{432 x ^{4} y ^{4} - 360 x ^{2} y ^{3} - 150 y ^{2}}{( 6 x ^{2} y - 5 ) ^{3}}

Expand the expression

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\frac{432 x ^{4} y ^{4} - 360 x ^{2} y ^{3} - 150 y ^{2}}{216 x ^{6} y ^{3} - 540 x ^{4} y ^{2} + 450 x ^{2} y - 125}

\frac{d ^{2} y}{d x ^{2}} = \frac{432 x ^{4} y ^{4} - 360 x ^{2} y ^{3} - 150 y ^{2}}{216 x ^{6} y ^{3} - 540 x ^{4} y ^{2} + 450 x ^{2} y - 125}

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