Solve x^2 - 4y^24x 8y^3=0

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = 128 y^{5}

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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 r = 4 \frac{cos ( θ )}{128 sin ^{5} ( θ )} r = - 4 \frac{cos ( θ )}{128 sin ^{5} ( θ )}

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{x - 64 y ^{5}}{320 x y ^{4}}

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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{48 y ^{5} x + 6144 y ^{10} - x ^{2}}{25600 y ^{9} x ^{2}}

Calculate

x^{2} - 4 y^{2} 4 x 8 y^{3} = 0

Simplify the expression

x^{2} - 128 y^{5} x = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} - 128 y^{5} x) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

2 x - 128 y^{5} - 640 x y^{4} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 x - 128 y^{5} - 640 x y^{4} \frac{d y}{d x} = 0

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

- 640 x y^{4} \frac{d y}{d x} = 0 - (2 x - 128 y^{5})

Subtract the terms

More Steps

- 640 x y^{4} \frac{d y}{d x} = - 2 x + 128 y^{5}

Divide both sides

\frac{- 640 x y ^{4} \frac{d y}{d x}}{- 640 x y ^{4}} = \frac{- 2 x + 128 y ^{5}}{- 640 x y ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x + 128 y ^{5}}{- 640 x y ^{4}}

Divide the numbers

More Steps

\frac{d y}{d x} = \frac{x - 64 y ^{5}}{320 x y ^{4}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{x - 64 y ^{5}}{320 x y ^{4}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{x - 64 y ^{5}}{320 x y ^{4}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( x - 64 y ^{5} ) \times 320 x y ^{4} - ( x - 64 y ^{5} ) \times \frac{d}{d x} ( 320 x y ^{4} )}{( 320 x y ^{4} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{( 1 - 320 y ^{4} \frac{d y}{d x} ) \times 320 x y ^{4} - ( x - 64 y ^{5} ) \times \frac{d}{d x} ( 320 x y ^{4} )}{( 320 x y ^{4} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{( 1 - 320 y ^{4} \frac{d y}{d x} ) \times 320 x y ^{4} - ( x - 64 y ^{5} ) ( 320 y ^{4} + 1280 x y ^{3} \frac{d y}{d x} )}{( 320 x y ^{4} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{320 x y ^{4} - 102400 y ^{8} x \frac{d y}{d x} - ( x - 64 y ^{5} ) ( 320 y ^{4} + 1280 x y ^{3} \frac{d y}{d x} )}{( 320 x y ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{320 x y ^{4} - 102400 y ^{8} x \frac{d y}{d x} - ( 320 x y ^{4} - 20480 y ^{9} + 1280 x ^{2} y ^{3} \frac{d y}{d x} - 81920 y ^{8} x \frac{d y}{d x} )}{( 320 x y ^{4} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 20480 y ^{8} x \frac{d y}{d x} + 20480 y ^{9} - 1280 x ^{2} y ^{3} \frac{d y}{d x}}{( 320 x y ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 20480 y ^{8} x \frac{d y}{d x} + 20480 y ^{9} - 1280 x ^{2} y ^{3} \frac{d y}{d x}}{32 0 ^{2} x ^{2} y ^{8}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{- 16 y ^{5} x \frac{d y}{d x} + 16 y ^{6} - x ^{2} \frac{d y}{d x}}{80 x ^{2} y ^{5}}

Use equation \frac{d y}{d x} = \frac{x - 64 y ^{5}}{320 x y ^{4}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- 16 y ^{5} x \times \frac{x - 64 y ^{5}}{320 x y ^{4}} + 16 y ^{6} - x ^{2} \times \frac{x - 64 y ^{5}}{320 x y ^{4}}}{80 x ^{2} y ^{5}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{48 y ^{5} x + 6144 y ^{10} - x ^{2}}{25600 y ^{9} x ^{2}}

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