Solve x^3 - 3xy^23y = 1

Question

Solve the equation

y = \frac{3 - 3 x ^{2} + 3 x ^{5}}{3 x}

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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{x ^{2} - 3 y ^{3}}{9 x y ^{2}}

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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{12 y ^{3} x ^{2} + 36 y ^{6} - 2 x ^{4}}{81 y ^{5} x ^{2}}

Calculate

x^{3} - 3 x y^{2} 3 y = 1

Simplify the expression

x^{3} - 9 x y^{3} = 1

Take the derivative of both sides

\frac{d}{d x} (x^{3} - 9 x y^{3}) = \frac{d}{d x} (1)

Calculate the derivative

More Steps

3 x^{2} - 9 y^{3} - 27 x y^{2} \frac{d y}{d x} = \frac{d}{d x} (1)

Calculate the derivative

3 x^{2} - 9 y^{3} - 27 x y^{2} \frac{d y}{d x} = 0

Move the expression to the right-hand side and change its sign

- 27 x y^{2} \frac{d y}{d x} = 0 - (3 x^{2} - 9 y^{3})

Subtract the terms

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

Divide both sides

\frac{- 27 x y ^{2} \frac{d y}{d x}}{- 27 x y ^{2}} = \frac{- 3 x ^{2} + 9 y ^{3}}{- 27 x y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{- 3 x ^{2} + 9 y ^{3}}{- 27 x y ^{2}}

Divide the numbers

More Steps

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

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

Calculate

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

Calculate

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

Calculate

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

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{y x ^{2} - 3 y ^{3} x \frac{d y}{d x} + 3 y ^{4} - 2 x ^{3} \frac{d y}{d x}}{9 x ^{2} y ^{3}}

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

\frac{d ^{2} y}{d x ^{2}} = \frac{y x ^{2} - 3 y ^{3} x \times \frac{x ^{2} - 3 y ^{3}}{9 x y ^{2}} + 3 y ^{4} - 2 x ^{3} \times \frac{x ^{2} - 3 y ^{3}}{9 x y ^{2}}}{9 x ^{2} y ^{3}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{12 y ^{3} x ^{2} + 36 y ^{6} - 2 x ^{4}}{81 y ^{5} x ^{2}}

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