Solve 32 x^3y^2 = (x y)^2

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = \frac{1}{32}

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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

Not symmetry with respect to the origin

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Rewrite the equation

r = 0 r = \frac{sec ( θ )}{32}

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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 - 48 y x}{32 x ^{2} - 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{- 160 y x + 3840 y x ^{2} + 2 y}{1024 x ^{4} - 64 x ^{3} + x ^{2}}

Calculate

32 x^{3} y^{2} = (x y)^{2}

Simplify the expression

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

Take the derivative of both sides

\frac{d}{d x} (32 x^{3} y^{2}) = \frac{d}{d x} (x^{2} y^{2})

Calculate the derivative

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96 x^{2} y^{2} + 64 x^{3} y \frac{d y}{d x} = \frac{d}{d x} (x^{2} y^{2})

Calculate the derivative

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96 x^{2} y^{2} + 64 x^{3} y \frac{d y}{d x} = 2 x y^{2} + 2 x^{2} y \frac{d y}{d x}

Move the expression to the left side

96 x^{2} y^{2} + 64 x^{3} y \frac{d y}{d x} - 2 x^{2} y \frac{d y}{d x} = 2 x y^{2}

Move the expression to the right side

64 x^{3} y \frac{d y}{d x} - 2 x^{2} y \frac{d y}{d x} = 2 x y^{2} - 96 x^{2} y^{2}

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

(64 x^{3} y - 2 x^{2} y) \frac{d y}{d x} = 2 x y^{2} - 96 x^{2} y^{2}

Divide both sides

\frac{( 64 x ^{3} y - 2 x ^{2} y ) \frac{d y}{d x}}{64 x ^{3} y - 2 x ^{2} y} = \frac{2 x y ^{2} - 96 x ^{2} y ^{2}}{64 x ^{3} y - 2 x ^{2} y}

Divide the numbers

\frac{d y}{d x} = \frac{2 x y ^{2} - 96 x ^{2} y ^{2}}{64 x ^{3} y - 2 x ^{2} y}

Divide the numbers

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\frac{d y}{d x} = \frac{y - 48 y x}{32 x ^{2} - x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{y - 48 y x}{32 x ^{2} - x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{y - 48 y x}{32 x ^{2} - x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( y - 48 y x ) \times ( 32 x ^{2} - x ) - ( y - 48 y x ) \times \frac{d}{d x} ( 32 x ^{2} - x )}{( 32 x ^{2} - x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( \frac{d y}{d x} - 48 y - 48 x \frac{d y}{d x} ) ( 32 x ^{2} - x ) - ( y - 48 y x ) \times \frac{d}{d x} ( 32 x ^{2} - x )}{( 32 x ^{2} - x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( \frac{d y}{d x} - 48 y - 48 x \frac{d y}{d x} ) ( 32 x ^{2} - x ) - ( y - 48 y x ) ( 64 x - 1 )}{( 32 x ^{2} - x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{80 x ^{2} \frac{d y}{d x} - x \frac{d y}{d x} - 1536 y x ^{2} + 48 y x - 1536 x ^{3} \frac{d y}{d x} - ( y - 48 y x ) ( 64 x - 1 )}{( 32 x ^{2} - x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{80 x ^{2} \frac{d y}{d x} - x \frac{d y}{d x} - 1536 y x ^{2} + 48 y x - 1536 x ^{3} \frac{d y}{d x} - ( 112 y x - 3072 y x ^{2} - y )}{( 32 x ^{2} - x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{80 x ^{2} \frac{d y}{d x} - x \frac{d y}{d x} + 1536 y x ^{2} - 64 y x - 1536 x ^{3} \frac{d y}{d x} + y}{( 32 x ^{2} - x ) ^{2}}

Use equation \frac{d y}{d x} = \frac{y - 48 y x}{32 x ^{2} - x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{80 x ^{2} \times \frac{y - 48 y x}{32 x ^{2} - x} - x \times \frac{y - 48 y x}{32 x ^{2} - x} + 1536 y x ^{2} - 64 y x - 1536 x ^{3} \times \frac{y - 48 y x}{32 x ^{2} - x} + y}{( 32 x ^{2} - x ) ^{2}}

Solution

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Calculate

\frac{80 x ^{2} \times \frac{y - 48 y x}{32 x ^{2} - x} - x \times \frac{y - 48 y x}{32 x ^{2} - x} + 1536 y x ^{2} - 64 y x - 1536 x ^{3} \times \frac{y - 48 y x}{32 x ^{2} - x} + y}{( 32 x ^{2} - x ) ^{2}}

Multiply the terms

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\frac{\frac{80 x ( y - 48 y x )}{32 x - 1} - x \times \frac{y - 48 y x}{32 x ^{2} - x} + 1536 y x ^{2} - 64 y x - 1536 x ^{3} \times \frac{y - 48 y x}{32 x ^{2} - x} + y}{( 32 x ^{2} - x ) ^{2}}

Multiply the terms

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\frac{\frac{80 x ( y - 48 y x )}{32 x - 1} - \frac{y - 48 y x}{32 x - 1} + 1536 y x ^{2} - 64 y x - 1536 x ^{3} \times \frac{y - 48 y x}{32 x ^{2} - x} + y}{( 32 x ^{2} - x ) ^{2}}

Multiply the terms

\frac{\frac{80 x ( y - 48 y x )}{32 x - 1} - \frac{y - 48 y x}{32 x - 1} + 1536 y x ^{2} - 64 y x - \frac{1536 x ^{2} ( y - 48 y x )}{32 x - 1} + y}{( 32 x ^{2} - x ) ^{2}}

Calculate the sum or difference

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\frac{- 160 y x + 3840 y x ^{2} + 2 y}{( 32 x ^{2} - x ) ^{2}}

Expand the expression

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\frac{- 160 y x + 3840 y x ^{2} + 2 y}{1024 x ^{4} - 64 x ^{3} + x ^{2}}

\frac{d ^{2} y}{d x ^{2}} = \frac{- 160 y x + 3840 y x ^{2} + 2 y}{1024 x ^{4} - 64 x ^{3} + x ^{2}}

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