Solve x^2 -2xy^3y^2=50

Question

Solve the equation

Solve for x

Solve for y

x = y^{5} + y^{10} + 50 x = y^{5} - y^{10} + 50

Evaluate

x^{2} - 2 x y^{3} \times y^{2} = 50

Multiply

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x^{2} - 2 x y^{5} = 50

Rewrite the expression

x^{2} - 2 y^{5} x = 50

Move the expression to the left side

x^{2} - 2 y^{5} x - 50 = 0

Substitute a= 1,b= - 2 y^{5} and c= - 50 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{2 y ^{5} \pm ( - 2 y ^{5} ) ^{2} - 4 ( - 50 )}{2}

Simplify the expression

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x = \frac{2 y ^{5} \pm 4 y ^{10} + 200}{2}

Simplify the radical expression

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x = \frac{2 y ^{5} \pm 2 y ^{10} + 50}{2}

Separate the equation into 2 possible cases

x = \frac{2 y ^{5} + 2 y ^{10} + 50}{2} x = \frac{2 y ^{5} - 2 y ^{10} + 50}{2}

Simplify the expression

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x = y^{5} + y^{10} + 50 x = \frac{2 y ^{5} - 2 y ^{10} + 50}{2}

Solution

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x = y^{5} + y^{10} + 50 x = y^{5} - y^{10} + 50

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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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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

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

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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{3 y ^{5} x + 6 y ^{10} - 4 x ^{2}}{25 y ^{9} x ^{2}}

Calculate

x^{2} - 2 x y^{3} y^{2} = 50

Simplify the expression

x^{2} - 2 x y^{5} = 50

Take the derivative of both sides

\frac{d}{d x} (x^{2} - 2 x y^{5}) = \frac{d}{d x} (50)

Calculate the derivative

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2 x - 2 y^{5} - 10 x y^{4} \frac{d y}{d x} = \frac{d}{d x} (50)

Calculate the derivative

2 x - 2 y^{5} - 10 x y^{4} \frac{d y}{d x} = 0

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

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

Subtract the terms

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

Divide both sides

\frac{- 10 x y ^{4} \frac{d y}{d x}}{- 10 x y ^{4}} = \frac{- 2 x + 2 y ^{5}}{- 10 x y ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x + 2 y ^{5}}{- 10 x y ^{4}}

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 1 - 5 y ^{4} \frac{d y}{d x} ) \times 5 x y ^{4} - ( x - y ^{5} ) ( 5 y ^{4} + 20 x y ^{3} \frac{d y}{d x} )}{( 5 x y ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{5 x y ^{4} - 25 y ^{8} x \frac{d y}{d x} - ( x - y ^{5} ) ( 5 y ^{4} + 20 x y ^{3} \frac{d y}{d x} )}{( 5 x y ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{5 x y ^{4} - 25 y ^{8} x \frac{d y}{d x} - ( 5 x y ^{4} - 5 y ^{9} + 20 x ^{2} y ^{3} \frac{d y}{d x} - 20 y ^{8} x \frac{d y}{d x} )}{( 5 x y ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{8} x \frac{d y}{d x} + 5 y ^{9} - 20 x ^{2} y ^{3} \frac{d y}{d x}}{( 5 x y ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{8} x \frac{d y}{d x} + 5 y ^{9} - 20 x ^{2} y ^{3} \frac{d y}{d x}}{25 x ^{2} y ^{8}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{- y ^{5} x \frac{d y}{d x} + y ^{6} - 4 x ^{2} \frac{d y}{d x}}{5 x ^{2} y ^{5}}

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

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

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{3 y ^{5} x + 6 y ^{10} - 4 x ^{2}}{25 y ^{9} x ^{2}}

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