Solve xy y^2=tan x y

Question

Solve the equation

y = 0 y = \frac{x tan ( x )}{∣ x ∣} y = - \frac{x tan ( x )}{∣ x ∣}

Evaluate

x y \times y^{2} = tan (x) \times y

Multiply

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Evaluate

x y \times y^{2}

Multiply the terms with the same base by adding their exponents

x y^{1 + 2}

Add the numbers

x y^{3}

x y^{3} = tan (x) \times y

Add or subtract both sides

x y^{3} - tan (x) \times y = 0

Factor the expression

y (x y^{2} - tan (x)) = 0

Separate the equation into 2 possible cases

y = 0 x y^{2} - tan (x) = 0

Solution

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Evaluate

x y^{2} - tan (x) = 0

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

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

Add the terms

x y^{2} = tan (x)

Divide both sides

\frac{x y ^{2}}{x} = \frac{tan ( x )}{x}

Divide the numbers

y^{2} = \frac{tan ( x )}{x}

Take the root of both sides of the equation and remember to use both positive and negative roots

y = \pm \frac{tan ( x )}{x}

Simplify the expression

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Evaluate

\frac{tan ( x )}{x}

Rewrite the expression

\frac{tan ( x ) \times x}{x \times x}

Use the commutative property to reorder the terms

\frac{x tan ( x )}{x \times x}

Calculate

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

To take a root of a fraction,take the root of the numerator and denominator separately

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

Simplify the radical expression

\frac{x tan ( x )}{∣ x ∣}

y = \pm \frac{x tan ( x )}{∣ x ∣}

Separate the equation into 2 possible cases

y = \frac{x tan ( x )}{∣ x ∣} y = - \frac{x tan ( x )}{∣ x ∣}

y = 0 y = \frac{x tan ( x )}{∣ x ∣} y = - \frac{x tan ( x )}{∣ x ∣}

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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

x y \times y^{2} = tan (x) \times y

Multiply

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Evaluate

x y \times y^{2}

Multiply the terms with the same base by adding their exponents

x y^{1 + 2}

Add the numbers

x y^{3}

x y^{3} = tan (x) \times y

Multiply the terms

x y^{3} = y tan (x)

To test if the graph of x y^{3} = y tan (x) is symmetry with respect to the origin,substitute -x for x and -y for y

- x (- y)^{3} = - y tan (- x)

Evaluate

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Evaluate

- x (- y)^{3}

Rewrite the expression

- x (- y^{3})

Multiplying or dividing an even number of negative terms equals a positive

x y^{3}

x y^{3} = - y tan (- x)

Evaluate

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Evaluate

- y tan (- x)

Use tan (- t) = - tan (t) to transform the expression

- y (- tan (x))

Rewrite the expression

y tan (x)

x y^{3} = y tan (x)

Solution

Symmetry with respect to the origin

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Find the first derivative

\frac{d y}{d x} = \frac{sec ^{2} ( x ) \times y - y ^{3}}{3 x y ^{2} - tan ( x )}

Calculate

x y y^{2} = tan (x) y

Simplify the expression

x y^{3} = y tan (x)

Take the derivative of both sides

\frac{d}{d x} (x y^{3}) = \frac{d}{d x} (y tan (x))

Calculate the derivative

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Evaluate

\frac{d}{d x} (x y^{3})

Use differentiation rules

\frac{d}{d x} (x) \times y^{3} + x \times \frac{d}{d x} (y^{3})

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

y^{3} + x \times \frac{d}{d x} (y^{3})

Evaluate the derivative

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Evaluate

\frac{d}{d x} (y^{3})

Use differentiation rules

\frac{d}{d y} (y^{3}) \times \frac{d y}{d x}

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

3 y^{2} \frac{d y}{d x}

y^{3} + 3 x y^{2} \frac{d y}{d x}

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

Calculate the derivative

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Evaluate

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

Use differentiation rules

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

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}

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

Use \frac{d}{d x} (tan x) = sec^{2} x to find derivative

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

y^{3} + 3 x y^{2} \frac{d y}{d x} = \frac{d y}{d x} \times tan (x) + sec^{2} (x) \times y

Rewrite the expression

y^{3} + 3 x y^{2} \frac{d y}{d x} = tan (x) \frac{d y}{d x} + sec^{2} (x) \times y

Move the expression to the left side

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

Move the expression to the right side

3 x y^{2} \frac{d y}{d x} - tan (x) \frac{d y}{d x} = sec^{2} (x) \times y - y^{3}

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

(3 x y^{2} - tan (x)) \frac{d y}{d x} = sec^{2} (x) \times y - y^{3}

Divide both sides

\frac{( 3 x y ^{2} - tan ( x ) ) \frac{d y}{d x}}{3 x y ^{2} - tan ( x )} = \frac{sec ^{2} ( x ) \times y - y ^{3}}{3 x y ^{2} - tan ( x )}

Solution

\frac{d y}{d x} = \frac{sec ^{2} ( x ) \times y - y ^{3}}{3 x y ^{2} - tan ( x )}

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