Solve 6x^2 xy-y^2=0

Question

Solve the equation

Solve for x

Solve for y

x = \frac{3 36 y}{6}

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

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r = 0 r = \frac{sin ( θ )}{6 cos ^{3} ( θ )} r = - \frac{sin ( θ )}{6 cos ^{3} ( θ )}

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{9 x ^{2} y}{3 x ^{3} - 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{324 x ^{7} y + 27 y ^{2} x ^{4} - 18 y ^{3} x}{27 x ^{9} - 27 x ^{6} y + 9 x ^{3} y ^{2} - y ^{3}}

Calculate

6 x^{2} x y - y^{2} = 0

Simplify the expression

6 x^{3} y - y^{2} = 0

Take the derivative of both sides

\frac{d}{d x} (6 x^{3} y - y^{2}) = \frac{d}{d x} (0)

Calculate the derivative

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18 x^{2} y + 6 x^{3} \frac{d y}{d x} - 2 y \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

18 x^{2} y + 6 x^{3} \frac{d y}{d x} - 2 y \frac{d y}{d x} = 0

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

18 x^{2} y + (6 x^{3} - 2 y) \frac{d y}{d x} = 0

Move the constant to the right side

(6 x^{3} - 2 y) \frac{d y}{d x} = 0 - 18 x^{2} y

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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\frac{d y}{d x} = - \frac{9 x ^{2} y}{3 x ^{3} - y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{9 x ^{2} y}{3 x ^{3} - y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{9 x ^{2} y}{3 x ^{3} - y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 9 x ^{2} y ) \times ( 3 x ^{3} - y ) - 9 x ^{2} y \times \frac{d}{d x} ( 3 x ^{3} - y )}{( 3 x ^{3} - y ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 18 x y + 9 x ^{2} \frac{d y}{d x} ) ( 3 x ^{3} - y ) - 9 x ^{2} y \times \frac{d}{d x} ( 3 x ^{3} - y )}{( 3 x ^{3} - y ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{( 18 x y + 9 x ^{2} \frac{d y}{d x} ) ( 3 x ^{3} - y ) - 9 x ^{2} y ( 9 x ^{2} - \frac{d y}{d x} )}{( 3 x ^{3} - y ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{54 x ^{4} y - 18 x y ^{2} + 27 x ^{5} \frac{d y}{d x} - 9 x ^{2} y \frac{d y}{d x} - 9 x ^{2} y ( 9 x ^{2} - \frac{d y}{d x} )}{( 3 x ^{3} - y ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{54 x ^{4} y - 18 x y ^{2} + 27 x ^{5} \frac{d y}{d x} - 9 x ^{2} y \frac{d y}{d x} - ( 81 x ^{4} y - 9 x ^{2} y \frac{d y}{d x} )}{( 3 x ^{3} - y ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{- 27 x ^{4} y - 18 x y ^{2} + 27 x ^{5} \frac{d y}{d x}}{( 3 x ^{3} - y ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{9 x ^{2} y}{3 x ^{3} - y} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 27 x ^{4} y - 18 x y ^{2} + 27 x ^{5} ( - \frac{9 x ^{2} y}{3 x ^{3} - y} )}{( 3 x ^{3} - y ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{324 x ^{7} y + 27 y ^{2} x ^{4} - 18 y ^{3} x}{27 x ^{9} - 27 x ^{6} y + 9 x ^{3} y ^{2} - y ^{3}}

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