Solve 3y^6x=xy - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = 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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Rewrite the equation

r = 0 r = \frac{csc ( θ )}{5 3}

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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 - 3 y ^{6}}{18 x y ^{5} - 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{- 12 y ^{6} - 324 y ^{11} - 2 y + 1134 y ^{16}}{5832 y ^{15} x ^{2} - 972 y ^{10} x ^{2} + 54 y ^{5} x ^{2} - x ^{2}}

Calculate

3 y^{6} x = x y

Take the derivative of both sides

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

Calculate the derivative

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3 y^{6} + 18 x y^{5} \frac{d y}{d x} = \frac{d}{d x} (x y)

Calculate the derivative

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3 y^{6} + 18 x y^{5} \frac{d y}{d x} = y + x \frac{d y}{d x}

Move the expression to the left side

3 y^{6} + 18 x y^{5} \frac{d y}{d x} - x \frac{d y}{d x} = y

Move the expression to the right side

18 x y^{5} \frac{d y}{d x} - x \frac{d y}{d x} = y - 3 y^{6}

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

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

Divide both sides

\frac{( 18 x y ^{5} - x ) \frac{d y}{d x}}{18 x y ^{5} - x} = \frac{y - 3 y ^{6}}{18 x y ^{5} - x}

Divide the numbers

\frac{d y}{d x} = \frac{y - 3 y ^{6}}{18 x y ^{5} - x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{y - 3 y ^{6}}{18 x y ^{5} - x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{y - 3 y ^{6}}{18 x y ^{5} - x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( y - 3 y ^{6} ) \times ( 18 x y ^{5} - x ) - ( y - 3 y ^{6} ) \times \frac{d}{d x} ( 18 x y ^{5} - x )}{( 18 x y ^{5} - x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( \frac{d y}{d x} - 18 y ^{5} \frac{d y}{d x} ) ( 18 x y ^{5} - x ) - ( y - 3 y ^{6} ) \times \frac{d}{d x} ( 18 x y ^{5} - x )}{( 18 x y ^{5} - x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( \frac{d y}{d x} - 18 y ^{5} \frac{d y}{d x} ) ( 18 x y ^{5} - x ) - ( y - 3 y ^{6} ) ( 18 y ^{5} + 90 x y ^{4} \frac{d y}{d x} - 1 )}{( 18 x y ^{5} - x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{36 x y ^{5} \frac{d y}{d x} - x \frac{d y}{d x} - 324 y ^{10} x \frac{d y}{d x} - ( y - 3 y ^{6} ) ( 18 y ^{5} + 90 x y ^{4} \frac{d y}{d x} - 1 )}{( 18 x y ^{5} - x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{36 x y ^{5} \frac{d y}{d x} - x \frac{d y}{d x} - 324 y ^{10} x \frac{d y}{d x} - ( 21 y ^{6} + 90 y ^{5} x \frac{d y}{d x} - 54 y ^{11} - 270 y ^{10} x \frac{d y}{d x} - y )}{( 18 x y ^{5} - x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 54 x y ^{5} \frac{d y}{d x} - x \frac{d y}{d x} - 54 y ^{10} x \frac{d y}{d x} - 21 y ^{6} + 54 y ^{11} + y}{( 18 x y ^{5} - x ) ^{2}}

Use equation \frac{d y}{d x} = \frac{y - 3 y ^{6}}{18 x y ^{5} - x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- 54 x y ^{5} \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - x \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - 54 y ^{10} x \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - 21 y ^{6} + 54 y ^{11} + y}{( 18 x y ^{5} - x ) ^{2}}

Solution

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Calculate

\frac{- 54 x y ^{5} \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - x \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - 54 y ^{10} x \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - 21 y ^{6} + 54 y ^{11} + y}{( 18 x y ^{5} - x ) ^{2}}

Multiply the terms

\frac{- \frac{54 y ^{5} ( y - 3 y ^{6} )}{18 y ^{5} - 1} - x \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - 54 y ^{10} x \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - 21 y ^{6} + 54 y ^{11} + y}{( 18 x y ^{5} - x ) ^{2}}

Multiply the terms

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\frac{- \frac{54 y ^{5} ( y - 3 y ^{6} )}{18 y ^{5} - 1} - \frac{y - 3 y ^{6}}{18 y ^{5} - 1} - 54 y ^{10} x \times \frac{y - 3 y ^{6}}{18 x y ^{5} - x} - 21 y ^{6} + 54 y ^{11} + y}{( 18 x y ^{5} - x ) ^{2}}

Multiply the terms

\frac{- \frac{54 y ^{5} ( y - 3 y ^{6} )}{18 y ^{5} - 1} - \frac{y - 3 y ^{6}}{18 y ^{5} - 1} - \frac{54 y ^{10} ( y - 3 y ^{6} )}{18 y ^{5} - 1} - 21 y ^{6} + 54 y ^{11} + y}{( 18 x y ^{5} - x ) ^{2}}

Calculate the sum or difference

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\frac{\frac{- 12 y ^{6} - 324 y ^{11} - 2 y + 1134 y ^{16}}{18 y ^{5} - 1}}{( 18 x y ^{5} - x ) ^{2}}

Multiply by the reciprocal

\frac{- 12 y ^{6} - 324 y ^{11} - 2 y + 1134 y ^{16}}{18 y ^{5} - 1} \times \frac{1}{( 18 x y ^{5} - x ) ^{2}}

Multiply the terms

\frac{- 12 y ^{6} - 324 y ^{11} - 2 y + 1134 y ^{16}}{( 18 y ^{5} - 1 ) ( 18 x y ^{5} - x ) ^{2}}

Expand the expression

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\frac{- 12 y ^{6} - 324 y ^{11} - 2 y + 1134 y ^{16}}{5832 y ^{15} x ^{2} - 972 y ^{10} x ^{2} + 54 y ^{5} x ^{2} - x ^{2}}

\frac{d ^{2} y}{d x ^{2}} = \frac{- 12 y ^{6} - 324 y ^{11} - 2 y + 1134 y ^{16}}{5832 y ^{15} x ^{2} - 972 y ^{10} x ^{2} + 54 y ^{5} x ^{2} - x ^{2}}

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