Solve 2x^3y = 12222 x y

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = 3679 x = - 3679

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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 = 3679 \times ∣ sec (θ) ∣ r = - 3679 \times ∣ sec (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{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x}

Calculate

2 x^{3} y = 12222 x y

Take the derivative of both sides

\frac{d}{d x} (2 x^{3} y) = \frac{d}{d x} (12222 x y)

Calculate the derivative

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6 x^{2} y + 2 x^{3} \frac{d y}{d x} = \frac{d}{d x} (12222 x y)

Calculate the derivative

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6 x^{2} y + 2 x^{3} \frac{d y}{d x} = 12222 y + 12222 x \frac{d y}{d x}

Move the expression to the left side

6 x^{2} y + 2 x^{3} \frac{d y}{d x} - 12222 x \frac{d y}{d x} = 12222 y

Move the expression to the right side

2 x^{3} \frac{d y}{d x} - 12222 x \frac{d y}{d x} = 12222 y - 6 x^{2} y

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

(2 x^{3} - 12222 x) \frac{d y}{d x} = 12222 y - 6 x^{2} y

Divide both sides

\frac{( 2 x ^{3} - 12222 x ) \frac{d y}{d x}}{2 x ^{3} - 12222 x} = \frac{12222 y - 6 x ^{2} y}{2 x ^{3} - 12222 x}

Divide the numbers

\frac{d y}{d x} = \frac{12222 y - 6 x ^{2} y}{2 x ^{3} - 12222 x}

Solution

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\frac{d y}{d x} = \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 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{- 36666 y x ^{2} + 12 y x ^{4} + 74688642 y}{x ^{6} - 12222 x ^{4} + 611 1 ^{2} x ^{2}}

Calculate

2 x^{3} y = 12222 x y

Take the derivative of both sides

\frac{d}{d x} (2 x^{3} y) = \frac{d}{d x} (12222 x y)

Calculate the derivative

More Steps

6 x^{2} y + 2 x^{3} \frac{d y}{d x} = \frac{d}{d x} (12222 x y)

Calculate the derivative

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6 x^{2} y + 2 x^{3} \frac{d y}{d x} = 12222 y + 12222 x \frac{d y}{d x}

Move the expression to the left side

6 x^{2} y + 2 x^{3} \frac{d y}{d x} - 12222 x \frac{d y}{d x} = 12222 y

Move the expression to the right side

2 x^{3} \frac{d y}{d x} - 12222 x \frac{d y}{d x} = 12222 y - 6 x^{2} y

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

(2 x^{3} - 12222 x) \frac{d y}{d x} = 12222 y - 6 x^{2} y

Divide both sides

\frac{( 2 x ^{3} - 12222 x ) \frac{d y}{d x}}{2 x ^{3} - 12222 x} = \frac{12222 y - 6 x ^{2} y}{2 x ^{3} - 12222 x}

Divide the numbers

\frac{d y}{d x} = \frac{12222 y - 6 x ^{2} y}{2 x ^{3} - 12222 x}

Divide the numbers

More Steps

\frac{d y}{d x} = \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 6111 y - 3 x ^{2} y ) \times ( x ^{3} - 6111 x ) - ( 6111 y - 3 x ^{2} y ) \times \frac{d}{d x} ( x ^{3} - 6111 x )}{( x ^{3} - 6111 x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 6111 \frac{d y}{d x} - 6 x y - 3 x ^{2} \frac{d y}{d x} ) ( x ^{3} - 6111 x ) - ( 6111 y - 3 x ^{2} y ) \times \frac{d}{d x} ( x ^{3} - 6111 x )}{( x ^{3} - 6111 x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 6111 \frac{d y}{d x} - 6 x y - 3 x ^{2} \frac{d y}{d x} ) ( x ^{3} - 6111 x ) - ( 6111 y - 3 x ^{2} y ) ( 3 x ^{2} - 6111 )}{( x ^{3} - 6111 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{24444 x ^{3} \frac{d y}{d x} - 37344321 x \frac{d y}{d x} - 6 x ^{4} y + 36666 x ^{2} y - 3 x ^{5} \frac{d y}{d x} - ( 6111 y - 3 x ^{2} y ) ( 3 x ^{2} - 6111 )}{( x ^{3} - 6111 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{24444 x ^{3} \frac{d y}{d x} - 37344321 x \frac{d y}{d x} - 6 x ^{4} y + 36666 x ^{2} y - 3 x ^{5} \frac{d y}{d x} - ( 36666 y x ^{2} - 9 x ^{4} y - 37344321 y )}{( x ^{3} - 6111 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{24444 x ^{3} \frac{d y}{d x} - 37344321 x \frac{d y}{d x} + 3 x ^{4} y - 3 x ^{5} \frac{d y}{d x} + 37344321 y}{( x ^{3} - 6111 x ) ^{2}}

Use equation \frac{d y}{d x} = \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{24444 x ^{3} \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} - 37344321 x \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} + 3 x ^{4} y - 3 x ^{5} \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} + 37344321 y}{( x ^{3} - 6111 x ) ^{2}}

Solution

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Calculate

\frac{24444 x ^{3} \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} - 37344321 x \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} + 3 x ^{4} y - 3 x ^{5} \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} + 37344321 y}{( x ^{3} - 6111 x ) ^{2}}

Multiply the terms

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\frac{\frac{24444 x ^{2} ( 6111 y - 3 x ^{2} y )}{x ^{2} - 6111} - 37344321 x \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} + 3 x ^{4} y - 3 x ^{5} \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} + 37344321 y}{( x ^{3} - 6111 x ) ^{2}}

Multiply the terms

\frac{\frac{24444 x ^{2} ( 6111 y - 3 x ^{2} y )}{x ^{2} - 6111} - \frac{37344321 ( 6111 y - 3 x ^{2} y )}{x ^{2} - 6111} + 3 x ^{4} y - 3 x ^{5} \times \frac{6111 y - 3 x ^{2} y}{x ^{3} - 6111 x} + 37344321 y}{( x ^{3} - 6111 x ) ^{2}}

Multiply the terms

\frac{\frac{24444 x ^{2} ( 6111 y - 3 x ^{2} y )}{x ^{2} - 6111} - \frac{37344321 ( 6111 y - 3 x ^{2} y )}{x ^{2} - 6111} + 3 x ^{4} y - \frac{3 x ^{4} ( 6111 y - 3 x ^{2} y )}{x ^{2} - 6111} + 37344321 y}{( x ^{3} - 6111 x ) ^{2}}

Calculate the sum or difference

More Steps

\frac{- 36666 y x ^{2} + 12 y x ^{4} + 74688642 y}{( x ^{3} - 6111 x ) ^{2}}

Expand the expression

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\frac{- 36666 y x ^{2} + 12 y x ^{4} + 74688642 y}{x ^{6} - 12222 x ^{4} + 611 1 ^{2} x ^{2}}

\frac{d ^{2} y}{d x ^{2}} = \frac{- 36666 y x ^{2} + 12 y x ^{4} + 74688642 y}{x ^{6} - 12222 x ^{4} + 611 1 ^{2} x ^{2}}

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