Solve sin xy cos(x y)=1

Question

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

\frac{d y}{d x} = - \frac{y}{x}

Calculate

sin (x y) cos (x y) = 1

Take the derivative of both sides

\frac{d}{d x} (sin (x y) cos (x y)) = \frac{d}{d x} (1)

Calculate the derivative

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(y + x \frac{d y}{d x}) cos^{2} (x y) - sin^{2} (x y) (y + x \frac{d y}{d x}) = \frac{d}{d x} (1)

Calculate the derivative

(y + x \frac{d y}{d x}) cos^{2} (x y) - sin^{2} (x y) (y + x \frac{d y}{d x}) = 0

Rewrite the expression

cos^{2} (x y) (y + x \frac{d y}{d x}) - sin^{2} (x y) (y + x \frac{d y}{d x}) = 0

Calculate

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cos^{2} (x y) \times y + cos^{2} (x y) \times x \frac{d y}{d x} - sin^{2} (x y) \times y - sin^{2} (x y) \times x \frac{d y}{d x} = 0

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

cos^{2} (x y) \times y - sin^{2} (x y) \times y + (cos^{2} (x y) \times x - sin^{2} (x y) \times x) \frac{d y}{d x} = 0

Move the constant to the right side

(cos^{2} (x y) \times x - sin^{2} (x y) \times x) \frac{d y}{d x} = 0 - (cos^{2} (x y) \times y - sin^{2} (x y) \times y)

Subtract the terms

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(cos^{2} (x y) \times x - sin^{2} (x y) \times x) \frac{d y}{d x} = - cos^{2} (x y) \times y + sin^{2} (x y) \times y

Divide both sides

\frac{( cos ^{2} ( x y ) \times x - sin ^{2} ( x y ) \times x ) \frac{d y}{d x}}{cos ^{2} ( x y ) \times x - sin ^{2} ( x y ) \times x} = \frac{- cos ^{2} ( x y ) \times y + sin ^{2} ( x y ) \times y}{cos ^{2} ( x y ) \times x - sin ^{2} ( x y ) \times x}

Divide the numbers

\frac{d y}{d x} = \frac{- cos ^{2} ( x y ) \times y + sin ^{2} ( x y ) \times y}{cos ^{2} ( x y ) \times x - sin ^{2} ( x y ) \times x}

Solution

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\frac{d y}{d x} = - \frac{y}{x}

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Find the second derivative

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

Calculate

sin (x y) cos (x y) = 1

Take the derivative of both sides

\frac{d}{d x} (sin (x y) cos (x y)) = \frac{d}{d x} (1)

Calculate the derivative

More Steps

(y + x \frac{d y}{d x}) cos^{2} (x y) - sin^{2} (x y) (y + x \frac{d y}{d x}) = \frac{d}{d x} (1)

Calculate the derivative

(y + x \frac{d y}{d x}) cos^{2} (x y) - sin^{2} (x y) (y + x \frac{d y}{d x}) = 0

Rewrite the expression

cos^{2} (x y) (y + x \frac{d y}{d x}) - sin^{2} (x y) (y + x \frac{d y}{d x}) = 0

Calculate

More Steps

cos^{2} (x y) \times y + cos^{2} (x y) \times x \frac{d y}{d x} - sin^{2} (x y) \times y - sin^{2} (x y) \times x \frac{d y}{d x} = 0

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

cos^{2} (x y) \times y - sin^{2} (x y) \times y + (cos^{2} (x y) \times x - sin^{2} (x y) \times x) \frac{d y}{d x} = 0

Move the constant to the right side

(cos^{2} (x y) \times x - sin^{2} (x y) \times x) \frac{d y}{d x} = 0 - (cos^{2} (x y) \times y - sin^{2} (x y) \times y)

Subtract the terms

More Steps

(cos^{2} (x y) \times x - sin^{2} (x y) \times x) \frac{d y}{d x} = - cos^{2} (x y) \times y + sin^{2} (x y) \times y

Divide both sides

\frac{( cos ^{2} ( x y ) \times x - sin ^{2} ( x y ) \times x ) \frac{d y}{d x}}{cos ^{2} ( x y ) \times x - sin ^{2} ( x y ) \times x} = \frac{- cos ^{2} ( x y ) \times y + sin ^{2} ( x y ) \times y}{cos ^{2} ( x y ) \times x - sin ^{2} ( x y ) \times x}

Divide the numbers

\frac{d y}{d x} = \frac{- cos ^{2} ( x y ) \times y + sin ^{2} ( x y ) \times y}{cos ^{2} ( x y ) \times x - sin ^{2} ( x y ) \times x}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{y}{x}

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

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

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times x - y \times 1}{x ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{x \frac{d y}{d x} - y \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

\frac{d ^{2} y}{d x ^{2}} = - \frac{x \frac{d y}{d x} - y}{x ^{2}}

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

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

Solution

More Steps

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

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