Solve (x^2+9^2-1) x^2y^2=0

Question

Solve the equation

Solve for x

Solve for y

x = 0

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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 = 4 - 5 sec^{2} (θ) r = - 4 - 5 sec^{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{2 x ^{2} y + 80 y}{x ^{3} + 80 x}

Calculate

(x^{2} + 9^{2} - 1) x^{2} y^{2} = 0

Simplify the expression

x^{2} (x^{2} + 80) y^{2} = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} (x^{2} + 80) y^{2}) = \frac{d}{d x} (0)

Calculate the derivative

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2 x^{3} y^{2} + 160 x y^{2} + 2 x^{3} y^{2} + 2 x^{4} y \frac{d y}{d x} + 160 x^{2} y \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 x^{3} y^{2} + 160 x y^{2} + 2 x^{3} y^{2} + 2 x^{4} y \frac{d y}{d x} + 160 x^{2} y \frac{d y}{d x} = 0

Simplify

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4 x^{3} y^{2} + 160 x y^{2} + (2 x^{4} y + 160 x^{2} y) \frac{d y}{d x} = 0

Move the constant to the right side

(2 x^{4} y + 160 x^{2} y) \frac{d y}{d x} = 0 - (4 x^{3} y^{2} + 160 x y^{2})

Subtract the terms

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(2 x^{4} y + 160 x^{2} y) \frac{d y}{d x} = - 4 x^{3} y^{2} - 160 x y^{2}

Divide both sides

\frac{( 2 x ^{4} y + 160 x ^{2} y ) \frac{d y}{d x}}{2 x ^{4} y + 160 x ^{2} y} = \frac{- 4 x ^{3} y ^{2} - 160 x y ^{2}}{2 x ^{4} y + 160 x ^{2} y}

Divide the numbers

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

Solution

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\frac{d y}{d x} = - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x}

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Find the second derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{6 x ^{4} y + 400 x ^{2} y + 12800 y}{x ^{6} + 160 x ^{4} + 6400 x ^{2}}

Calculate

(x^{2} + 9^{2} - 1) x^{2} y^{2} = 0

Simplify the expression

x^{2} (x^{2} + 80) y^{2} = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} (x^{2} + 80) y^{2}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

2 x^{3} y^{2} + 160 x y^{2} + 2 x^{3} y^{2} + 2 x^{4} y \frac{d y}{d x} + 160 x^{2} y \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 x^{3} y^{2} + 160 x y^{2} + 2 x^{3} y^{2} + 2 x^{4} y \frac{d y}{d x} + 160 x^{2} y \frac{d y}{d x} = 0

Simplify

More Steps

4 x^{3} y^{2} + 160 x y^{2} + (2 x^{4} y + 160 x^{2} y) \frac{d y}{d x} = 0

Move the constant to the right side

(2 x^{4} y + 160 x^{2} y) \frac{d y}{d x} = 0 - (4 x^{3} y^{2} + 160 x y^{2})

Subtract the terms

More Steps

(2 x^{4} y + 160 x^{2} y) \frac{d y}{d x} = - 4 x^{3} y^{2} - 160 x y^{2}

Divide both sides

\frac{( 2 x ^{4} y + 160 x ^{2} y ) \frac{d y}{d x}}{2 x ^{4} y + 160 x ^{2} y} = \frac{- 4 x ^{3} y ^{2} - 160 x y ^{2}}{2 x ^{4} y + 160 x ^{2} y}

Divide the numbers

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

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x}

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 4 x y + 2 x ^{2} \frac{d y}{d x} + 80 \frac{d y}{d x} ) ( x ^{3} + 80 x ) - ( 2 x ^{2} y + 80 y ) \times \frac{d}{d x} ( x ^{3} + 80 x )}{( x ^{3} + 80 x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{( 4 x y + 2 x ^{2} \frac{d y}{d x} + 80 \frac{d y}{d x} ) ( x ^{3} + 80 x ) - ( 2 x ^{2} y + 80 y ) ( 3 x ^{2} + 80 )}{( x ^{3} + 80 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{4 x ^{4} y + 320 x ^{2} y + 2 x ^{5} \frac{d y}{d x} + 240 x ^{3} \frac{d y}{d x} + 6400 x \frac{d y}{d x} - ( 2 x ^{2} y + 80 y ) ( 3 x ^{2} + 80 )}{( x ^{3} + 80 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{4 x ^{4} y + 320 x ^{2} y + 2 x ^{5} \frac{d y}{d x} + 240 x ^{3} \frac{d y}{d x} + 6400 x \frac{d y}{d x} - ( 6 x ^{4} y + 400 y x ^{2} + 6400 y )}{( x ^{3} + 80 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{- 2 x ^{4} y - 80 x ^{2} y + 2 x ^{5} \frac{d y}{d x} + 240 x ^{3} \frac{d y}{d x} + 6400 x \frac{d y}{d x} - 6400 y}{( x ^{3} + 80 x ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 2 x ^{4} y - 80 x ^{2} y + 2 x ^{5} ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) + 240 x ^{3} ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) + 6400 x ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) - 6400 y}{( x ^{3} + 80 x ) ^{2}}

Solution

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Calculate

- \frac{- 2 x ^{4} y - 80 x ^{2} y + 2 x ^{5} ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) + 240 x ^{3} ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) + 6400 x ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) - 6400 y}{( x ^{3} + 80 x ) ^{2}}

Multiply

More Steps

- \frac{- 2 x ^{4} y - 80 x ^{2} y - \frac{2 x ^{4} ( 2 x ^{2} y + 80 y )}{x ^{2} + 80} + 240 x ^{3} ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) + 6400 x ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) - 6400 y}{( x ^{3} + 80 x ) ^{2}}

Multiply

More Steps

- \frac{- 2 x ^{4} y - 80 x ^{2} y - \frac{2 x ^{4} ( 2 x ^{2} y + 80 y )}{x ^{2} + 80} - \frac{240 x ^{2} ( 2 x ^{2} y + 80 y )}{x ^{2} + 80} + 6400 x ( - \frac{2 x ^{2} y + 80 y}{x ^{3} + 80 x} ) - 6400 y}{( x ^{3} + 80 x ) ^{2}}

Multiply

More Steps

- \frac{- 2 x ^{4} y - 80 x ^{2} y - \frac{2 x ^{4} ( 2 x ^{2} y + 80 y )}{x ^{2} + 80} - \frac{240 x ^{2} ( 2 x ^{2} y + 80 y )}{x ^{2} + 80} - \frac{6400 ( 2 x ^{2} y + 80 y )}{x ^{2} + 80} - 6400 y}{( x ^{3} + 80 x ) ^{2}}

Subtract the terms

More Steps

- \frac{- 6 x ^{4} y - 400 x ^{2} y - 12800 y}{( x ^{3} + 80 x ) ^{2}}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- (- \frac{6 x ^{4} y + 400 x ^{2} y + 12800 y}{( x ^{3} + 80 x ) ^{2}})

Calculate

\frac{6 x ^{4} y + 400 x ^{2} y + 12800 y}{( x ^{3} + 80 x ) ^{2}}

Expand the expression

More Steps

\frac{6 x ^{4} y + 400 x ^{2} y + 12800 y}{x ^{6} + 160 x ^{4} + 6400 x ^{2}}

\frac{d ^{2} y}{d x ^{2}} = \frac{6 x ^{4} y + 400 x ^{2} y + 12800 y}{x ^{6} + 160 x ^{4} + 6400 x ^{2}}

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