Solve x^22xy y^2=(x y)2x^22xy y^2

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = \frac{1}{2 y}

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Testing for symmetry

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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Rewrite the equation

r = 0 r = csc (2 θ) r = - csc (2 θ)

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

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

Calculate

x^{2} 2 x y y^{2} = (x y) 2 x^{2} 2 x y y^{2}

Simplify the expression

2 x^{3} y^{3} = 4 x^{4} y^{4}

Take the derivative of both sides

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

Calculate the derivative

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

Calculate the derivative

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6 x^{2} y^{3} + 6 x^{3} y^{2} \frac{d y}{d x} = 16 x^{3} y^{4} + 16 x^{4} y^{3} \frac{d y}{d x}

Move the expression to the left side

6 x^{2} y^{3} + 6 x^{3} y^{2} \frac{d y}{d x} - 16 x^{4} y^{3} \frac{d y}{d x} = 16 x^{3} y^{4}

Move the expression to the right side

6 x^{3} y^{2} \frac{d y}{d x} - 16 x^{4} y^{3} \frac{d y}{d x} = 16 x^{3} y^{4} - 6 x^{2} y^{3}

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

(6 x^{3} y^{2} - 16 x^{4} y^{3}) \frac{d y}{d x} = 16 x^{3} y^{4} - 6 x^{2} y^{3}

Divide both sides

\frac{( 6 x ^{3} y ^{2} - 16 x ^{4} y ^{3} ) \frac{d y}{d x}}{6 x ^{3} y ^{2} - 16 x ^{4} y ^{3}} = \frac{16 x ^{3} y ^{4} - 6 x ^{2} y ^{3}}{6 x ^{3} y ^{2} - 16 x ^{4} y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{16 x ^{3} y ^{4} - 6 x ^{2} y ^{3}}{6 x ^{3} y ^{2} - 16 x ^{4} y ^{3}}

Solution

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

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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{2 y}{x ^{2}}

Calculate

x^{2} 2 x y y^{2} = (x y) 2 x^{2} 2 x y y^{2}

Simplify the expression

2 x^{3} y^{3} = 4 x^{4} y^{4}

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

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6 x^{2} y^{3} + 6 x^{3} y^{2} \frac{d y}{d x} = 16 x^{3} y^{4} + 16 x^{4} y^{3} \frac{d y}{d x}

Move the expression to the left side

6 x^{2} y^{3} + 6 x^{3} y^{2} \frac{d y}{d x} - 16 x^{4} y^{3} \frac{d y}{d x} = 16 x^{3} y^{4}

Move the expression to the right side

6 x^{3} y^{2} \frac{d y}{d x} - 16 x^{4} y^{3} \frac{d y}{d x} = 16 x^{3} y^{4} - 6 x^{2} y^{3}

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

(6 x^{3} y^{2} - 16 x^{4} y^{3}) \frac{d y}{d x} = 16 x^{3} y^{4} - 6 x^{2} y^{3}

Divide both sides

\frac{( 6 x ^{3} y ^{2} - 16 x ^{4} y ^{3} ) \frac{d y}{d x}}{6 x ^{3} y ^{2} - 16 x ^{4} y ^{3}} = \frac{16 x ^{3} y ^{4} - 6 x ^{2} y ^{3}}{6 x ^{3} y ^{2} - 16 x ^{4} y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{16 x ^{3} y ^{4} - 6 x ^{2} y ^{3}}{6 x ^{3} y ^{2} - 16 x ^{4} y ^{3}}

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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