Solve x^2 y^26x - 4y - 12 = 0

Question

Solve the equation

Solve for x

Solve for y

x = \frac{3 18 y ^{2} + 54 y}{3 y}

Show Solution

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

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2}

Show Solution

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{405 x ^{7} y ^{4} - 108 x ^{4} y ^{3} - 36 x y ^{2}}{108 x ^{9} y ^{3} - 108 x ^{6} y ^{2} + 36 x ^{3} y - 4}

Calculate

x^{2} y^{2} 6 x - 4 y - 12 = 0

Simplify the expression

6 x^{3} y^{2} - 4 y - 12 = 0

Take the derivative of both sides

\frac{d}{d x} (6 x^{3} y^{2} - 4 y - 12) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

18 x^{2} y^{2} + 12 x^{3} y \frac{d y}{d x} - 4 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

18 x^{2} y^{2} + 12 x^{3} y \frac{d y}{d x} - 4 \frac{d y}{d x} = 0

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

18 x^{2} y^{2} + (12 x^{3} y - 4) \frac{d y}{d x} = 0

Move the constant to the right side

(12 x^{3} y - 4) \frac{d y}{d x} = 0 - 18 x^{2} y^{2}

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

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

Divide both sides

\frac{( 12 x ^{3} y - 4 ) \frac{d y}{d x}}{12 x ^{3} y - 4} = \frac{- 18 x ^{2} y ^{2}}{12 x ^{3} y - 4}

Divide the numbers

\frac{d y}{d x} = \frac{- 18 x ^{2} y ^{2}}{12 x ^{3} y - 4}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 9 x ^{2} y ^{2} ) \times ( 6 x ^{3} y - 2 ) - 9 x ^{2} y ^{2} \times \frac{d}{d x} ( 6 x ^{3} y - 2 )}{( 6 x ^{3} y - 2 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 18 x y ^{2} + 18 x ^{2} y \frac{d y}{d x} ) ( 6 x ^{3} y - 2 ) - 9 x ^{2} y ^{2} \times \frac{d}{d x} ( 6 x ^{3} y - 2 )}{( 6 x ^{3} y - 2 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 18 x y ^{2} + 18 x ^{2} y \frac{d y}{d x} ) ( 6 x ^{3} y - 2 ) - 9 x ^{2} y ^{2} ( 18 x ^{2} y + 6 x ^{3} \frac{d y}{d x} )}{( 6 x ^{3} y - 2 ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{108 x ^{4} y ^{3} - 36 x y ^{2} + 108 x ^{5} y ^{2} \frac{d y}{d x} - 36 x ^{2} y \frac{d y}{d x} - 9 x ^{2} y ^{2} ( 18 x ^{2} y + 6 x ^{3} \frac{d y}{d x} )}{( 6 x ^{3} y - 2 ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{108 x ^{4} y ^{3} - 36 x y ^{2} + 108 x ^{5} y ^{2} \frac{d y}{d x} - 36 x ^{2} y \frac{d y}{d x} - ( 162 x ^{4} y ^{3} + 54 x ^{5} y ^{2} \frac{d y}{d x} )}{( 6 x ^{3} y - 2 ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 54 x ^{4} y ^{3} - 36 x y ^{2} + 54 x ^{5} y ^{2} \frac{d y}{d x} - 36 x ^{2} y \frac{d y}{d x}}{( 6 x ^{3} y - 2 ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 27 x ^{4} y ^{3} - 18 x y ^{2} + 27 x ^{5} y ^{2} \frac{d y}{d x} - 18 x ^{2} y \frac{d y}{d x}}{2 ( 3 x ^{3} y - 1 ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 27 x ^{4} y ^{3} - 18 x y ^{2} + 27 x ^{5} y ^{2} ( - \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2} ) - 18 x ^{2} y ( - \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2} )}{2 ( 3 x ^{3} y - 1 ) ^{2}}

Solution

More Steps

Calculate

- \frac{- 27 x ^{4} y ^{3} - 18 x y ^{2} + 27 x ^{5} y ^{2} ( - \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2} ) - 18 x ^{2} y ( - \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2} )}{2 ( 3 x ^{3} y - 1 ) ^{2}}

Multiply

More Steps

- \frac{- 27 x ^{4} y ^{3} - 18 x y ^{2} - \frac{243 x ^{7} y ^{4}}{6 x ^{3} y - 2} - 18 x ^{2} y ( - \frac{9 x ^{2} y ^{2}}{6 x ^{3} y - 2} )}{2 ( 3 x ^{3} y - 1 ) ^{2}}

Multiply

More Steps

- \frac{- 27 x ^{4} y ^{3} - 18 x y ^{2} - \frac{243 x ^{7} y ^{4}}{6 x ^{3} y - 2} + \frac{81 x ^{4} y ^{3}}{3 x ^{3} y - 1}}{2 ( 3 x ^{3} y - 1 ) ^{2}}

Calculate the sum or difference

More Steps

- \frac{\frac{- 405 x ^{7} y ^{4} + 108 x ^{4} y ^{3} + 36 x y ^{2}}{6 x ^{3} y - 2}}{2 ( 3 x ^{3} y - 1 ) ^{2}}

Divide the terms

More Steps

- \frac{- 405 x ^{7} y ^{4} + 108 x ^{4} y ^{3} + 36 x y ^{2}}{2 ( 6 x ^{3} y - 2 ) ( 3 x ^{3} y - 1 ) ^{2}}

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

\frac{405 x ^{7} y ^{4} - 108 x ^{4} y ^{3} - 36 x y ^{2}}{2 ( 6 x ^{3} y - 2 ) ( 3 x ^{3} y - 1 ) ^{2}}

Expand the expression

More Steps

\frac{405 x ^{7} y ^{4} - 108 x ^{4} y ^{3} - 36 x y ^{2}}{108 x ^{9} y ^{3} - 108 x ^{6} y ^{2} + 36 x ^{3} y - 4}

\frac{d ^{2} y}{d x ^{2}} = \frac{405 x ^{7} y ^{4} - 108 x ^{4} y ^{3} - 36 x y ^{2}}{108 x ^{9} y ^{3} - 108 x ^{6} y ^{2} + 36 x ^{3} y - 4}

Show Solution