Solve x y=8 x-y^4

Question

Solve the equation

x = - \frac{y ^{4}}{y - 8}

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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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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{8 - y}{x + 4 y ^{3}}

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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{- 16 x + 2 x y + 128 y ^{3} - 4 y ^{4} - 768 y ^{2}}{x ^{3} + 12 x ^{2} y ^{3} + 48 x y ^{6} + 64 y ^{9}}

Calculate

x y = 8 x - y^{4}

Take the derivative of both sides

\frac{d}{d x} (x y) = \frac{d}{d x} (8 x - y^{4})

Calculate the derivative

More Steps

y + x \frac{d y}{d x} = \frac{d}{d x} (8 x - y^{4})

Calculate the derivative

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y + x \frac{d y}{d x} = 8 - 4 y^{3} \frac{d y}{d x}

Move the expression to the left side

y + x \frac{d y}{d x} + 4 y^{3} \frac{d y}{d x} = 8

Move the expression to the right side

x \frac{d y}{d x} + 4 y^{3} \frac{d y}{d x} = 8 - y

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

(x + 4 y^{3}) \frac{d y}{d x} = 8 - y

Divide both sides

\frac{( x + 4 y ^{3} ) \frac{d y}{d x}}{x + 4 y ^{3}} = \frac{8 - y}{x + 4 y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{8 - y}{x + 4 y ^{3}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{8 - y}{x + 4 y ^{3}})

Calculate the derivative

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

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 8 - y ) \times ( x + 4 y ^{3} ) - ( 8 - y ) \times \frac{d}{d x} ( x + 4 y ^{3} )}{( x + 4 y ^{3} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- \frac{d y}{d x} \times ( x + 4 y ^{3} ) - ( 8 - y ) \times \frac{d}{d x} ( x + 4 y ^{3} )}{( x + 4 y ^{3} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- \frac{d y}{d x} \times ( x + 4 y ^{3} ) - ( 8 - y ) ( 1 + 12 y ^{2} \frac{d y}{d x} )}{( x + 4 y ^{3} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- x \frac{d y}{d x} - 4 y ^{3} \frac{d y}{d x} - ( 8 - y ) ( 1 + 12 y ^{2} \frac{d y}{d x} )}{( x + 4 y ^{3} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- x \frac{d y}{d x} - 4 y ^{3} \frac{d y}{d x} - ( 8 - y + 96 y ^{2} \frac{d y}{d x} - 12 y ^{3} \frac{d y}{d x} )}{( x + 4 y ^{3} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- x \frac{d y}{d x} + 8 y ^{3} \frac{d y}{d x} - 8 + y - 96 y ^{2} \frac{d y}{d x}}{( x + 4 y ^{3} ) ^{2}}

Use equation \frac{d y}{d x} = \frac{8 - y}{x + 4 y ^{3}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- x \times \frac{8 - y}{x + 4 y ^{3}} + 8 y ^{3} \times \frac{8 - y}{x + 4 y ^{3}} - 8 + y - 96 y ^{2} \times \frac{8 - y}{x + 4 y ^{3}}}{( x + 4 y ^{3} ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 16 x + 2 x y + 128 y ^{3} - 4 y ^{4} - 768 y ^{2}}{x ^{3} + 12 x ^{2} y ^{3} + 48 x y ^{6} + 64 y ^{9}}

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