Solve x^2 y^2 xy=0

Solve the equation

Solve for x

Solve for y

x = 0

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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 θ = \frac{kπ}{2}, k \in Z

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{y}{x}

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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}{x ^{2}}

Calculate

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

Simplify the expression

x^{3} y^{3} = 0

Take the derivative of both sides

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

Calculate the derivative

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

Calculate the derivative

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

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

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

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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\frac{d y}{d x} = - \frac{y}{x}

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

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

Use the commutative property to reorder the terms

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

Any expression multiplied by 1 remains the same

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

Use equation \frac{d y}{d x} = - \frac{y}{x} to substitute

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

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{2 y}{x ^{2}}

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