Solve x-y^2xy=3 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = \frac{3}{1 - y ^{3}}

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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{1 - y ^{3}}{3 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{- 2 y ^{3} + 4 y ^{6} - 2}{9 y ^{5} x ^{2}}

Calculate

x - y^{2} x y = 3

Simplify the expression

x - y^{3} x = 3

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

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

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

- 3 x y^{2} \frac{d y}{d x} = 0 - (1 - y^{3})

Subtract the terms

More Steps

- 3 x y^{2} \frac{d y}{d x} = - 1 + y^{3}

Divide both sides

\frac{- 3 x y ^{2} \frac{d y}{d x}}{- 3 x y ^{2}} = \frac{- 1 + y ^{3}}{- 3 x y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{- 1 + y ^{3}}{- 3 x y ^{2}}

Divide the numbers

More Steps

\frac{d y}{d x} = \frac{1 - y ^{3}}{3 x y ^{2}}

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 3 y ^{2} \frac{d y}{d x} \times 3 x y ^{2} - ( 1 - y ^{3} ) ( 3 y ^{2} + 6 x y \frac{d y}{d x} )}{( 3 x y ^{2} ) ^{2}}

Calculate

More Steps

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

Calculate

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

Calculate

More Steps

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

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 3 y ^{4} x \frac{d y}{d x} - 3 y ^{2} + 3 y ^{5} - 6 x y \frac{d y}{d x}}{9 x ^{2} y ^{4}}

Calculate

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

Use equation \frac{d y}{d x} = \frac{1 - y ^{3}}{3 x y ^{2}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- y ^{3} x \times \frac{1 - y ^{3}}{3 x y ^{2}} - y + y ^{4} - 2 x \times \frac{1 - y ^{3}}{3 x y ^{2}}}{3 x ^{2} y ^{3}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 2 y ^{3} + 4 y ^{6} - 2}{9 y ^{5} x ^{2}}

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