Solve 2x^2 y = x-2xy^6

Question

Solve the equation

x = 0 x = \frac{1 - 2 y ^{6}}{2 y}

Evaluate

2 x^{2} y = x - 2 x y^{6}

Rewrite the expression

2 y x^{2} = x - 2 y^{6} x

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

2 y x^{2} = (1 - 2 y^{6}) x

Add or subtract both sides

2 y x^{2} - (1 - 2 y^{6}) x = 0

Calculate

2 y x^{2} + (- 1 + 2 y^{6}) x = 0

Expand the expression

2 y x^{2} - x + 2 y^{6} x = 0

Factor the expression

More Steps

Evaluate

2 y x^{2} - x + 2 y^{6} x

Rewrite the expression

x \times 2 y x - x + x \times 2 y^{6}

Factor out x from the expression

x (2 y x - 1 + 2 y^{6})

x (2 y x - 1 + 2 y^{6}) = 0

When the product of factors equals 0,at least one factor is 0

x = 0 2 y x - 1 + 2 y^{6} = 0

Solution

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Evaluate

2 y x - 1 + 2 y^{6} = 0

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

2 y x = 0 - (- 1 + 2 y^{6})

Subtract the terms

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Evaluate

0 - (- 1 + 2 y^{6})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

0 + 1 - 2 y^{6}

Removing 0 doesn't change the value,so remove it from the expression

1 - 2 y^{6}

2 y x = 1 - 2 y^{6}

Divide both sides

\frac{2 y x}{2 y} = \frac{1 - 2 y ^{6}}{2 y}

Divide the numbers

x = \frac{1 - 2 y ^{6}}{2 y}

x = 0 x = \frac{1 - 2 y ^{6}}{2 y}

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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

Evaluate

2 x^{2} y = x - 2 x y^{6}

To test if the graph of 2 x^{2} y = x - 2 x y^{6} is symmetry with respect to the origin,substitute -x for x and -y for y

2 (- x)^{2} (- y) = - x - 2 (- x) (- y)^{6}

Evaluate

More Steps

Evaluate

2 (- x)^{2} (- y)

Any expression multiplied by 1 remains the same

- 2 (- x)^{2} y

Multiply the terms

- 2 x^{2} y

- 2 x^{2} y = - x - 2 (- x) (- y)^{6}

Evaluate

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Evaluate

- x - 2 (- x) (- y)^{6}

Multiply

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Multiply the terms

2 (- x) (- y)^{6}

Any expression multiplied by 1 remains the same

- 2 x (- y)^{6}

Multiply the terms

- 2 x y^{6}

- x - (- 2 x y^{6})

Rewrite the expression

- x + 2 x y^{6}

- 2 x^{2} y = - x + 2 x y^{6}

Solution

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{1 - 2 y ^{6} - 4 x y}{2 x ^{2} + 12 x y ^{5}}

Calculate

2 x^{2} y = x - 2 x y^{6}

Take the derivative of both sides

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

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 x^{2} y)

Use differentiation rules

\frac{d}{d x} (2 x^{2}) \times y + 2 x^{2} \times \frac{d}{d x} (y)

Evaluate the derivative

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Evaluate

\frac{d}{d x} (2 x^{2})

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

2 \times \frac{d}{d x} (x^{2})

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

2 \times 2 x

Multiply the terms

4 x

4 x y + 2 x^{2} \times \frac{d}{d x} (y)

Evaluate the derivative

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Evaluate

\frac{d}{d x} (y)

Use differentiation rules

\frac{d}{d y} (y) \times \frac{d y}{d x}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{d y}{d x}

4 x y + 2 x^{2} \frac{d y}{d x}

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

Calculate the derivative

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Evaluate

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

Use differentiation rules

\frac{d}{d x} (x) + \frac{d}{d x} (- 2 x y^{6})

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

1 + \frac{d}{d x} (- 2 x y^{6})

Evaluate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

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

Evaluate the derivative

- 2 y^{6} - 12 x y^{5} \frac{d y}{d x}

1 - 2 y^{6} - 12 x y^{5} \frac{d y}{d x}

4 x y + 2 x^{2} \frac{d y}{d x} = 1 - 2 y^{6} - 12 x y^{5} \frac{d y}{d x}

Move the expression to the left side

4 x y + 2 x^{2} \frac{d y}{d x} + 12 x y^{5} \frac{d y}{d x} = 1 - 2 y^{6}

Move the expression to the right side

2 x^{2} \frac{d y}{d x} + 12 x y^{5} \frac{d y}{d x} = 1 - 2 y^{6} - 4 x y

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

(2 x^{2} + 12 x y^{5}) \frac{d y}{d x} = 1 - 2 y^{6} - 4 x y

Divide both sides

\frac{( 2 x ^{2} + 12 x y ^{5} ) \frac{d y}{d x}}{2 x ^{2} + 12 x y ^{5}} = \frac{1 - 2 y ^{6} - 4 x y}{2 x ^{2} + 12 x y ^{5}}

Solution

\frac{d y}{d x} = \frac{1 - 2 y ^{6} - 4 x y}{2 x ^{2} + 12 x y ^{5}}

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