Solve x^2 y^25x^2y^260=37xy

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = \frac{3 37 \times 30 0 ^{2}}{300 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

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Rewrite the equation

r = 0 r = \frac{6 37 \times 30 0 ^{5} csc ^{3} ( θ ) sec ^{3} ( θ )}{300} r = - \frac{6 37 \times 30 0 ^{5} csc ^{3} ( θ ) sec ^{3} ( θ )}{300}

Evaluate

x^{2} y^{2} \times 5 x^{2} y^{2} \times 60 = 37 x y

Evaluate

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300 x^{4} y^{4} = 37 x y

Move the expression to the left side

300 x^{4} y^{4} - 37 x y = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

300 (cos (θ) \times r)^{4} (sin (θ) \times r)^{4} - 37 cos (θ) \times r sin (θ) \times r = 0

Factor the expression

300 cos^{4} (θ) sin^{4} (θ) \times r^{8} - 37 cos (θ) sin (θ) \times r^{2} = 0

Simplify the expression

300 cos^{4} (θ) sin^{4} (θ) \times r^{8} - \frac{37}{2} sin (2 θ) \times r^{2} = 0

Factor the expression

r^{2} (300 cos^{4} (θ) sin^{4} (θ) \times r^{6} - \frac{37}{2} sin (2 θ)) = 0

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

r^{2} = 0 300 cos^{4} (θ) sin^{4} (θ) \times r^{6} - \frac{37}{2} sin (2 θ) = 0

Evaluate

r = 0 300 cos^{4} (θ) sin^{4} (θ) \times r^{6} - \frac{37}{2} sin (2 θ) = 0

Solution

More Steps

r = 0 r = \frac{6 37 \times 30 0 ^{5} csc ^{3} ( θ ) sec ^{3} ( θ )}{300} r = - \frac{6 37 \times 30 0 ^{5} csc ^{3} ( θ ) sec ^{3} ( θ )}{300}

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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{y}{x}

Calculate

x^{2} y^{2} 5 x^{2} y^{2} 60 = 37 x y

Simplify the expression

300 x^{4} y^{4} = 37 x y

Take the derivative of both sides

\frac{d}{d x} (300 x^{4} y^{4}) = \frac{d}{d x} (37 x y)

Calculate the derivative

More Steps

1200 x^{3} y^{4} + 1200 x^{4} y^{3} \frac{d y}{d x} = \frac{d}{d x} (37 x y)

Calculate the derivative

More Steps

1200 x^{3} y^{4} + 1200 x^{4} y^{3} \frac{d y}{d x} = 37 y + 37 x \frac{d y}{d x}

Move the expression to the left side

1200 x^{3} y^{4} + 1200 x^{4} y^{3} \frac{d y}{d x} - 37 x \frac{d y}{d x} = 37 y

Move the expression to the right side

1200 x^{4} y^{3} \frac{d y}{d x} - 37 x \frac{d y}{d x} = 37 y - 1200 x^{3} y^{4}

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

(1200 x^{4} y^{3} - 37 x) \frac{d y}{d x} = 37 y - 1200 x^{3} y^{4}

Divide both sides

\frac{( 1200 x ^{4} y ^{3} - 37 x ) \frac{d y}{d x}}{1200 x ^{4} y ^{3} - 37 x} = \frac{37 y - 1200 x ^{3} y ^{4}}{1200 x ^{4} y ^{3} - 37 x}

Divide the numbers

\frac{d y}{d x} = \frac{37 y - 1200 x ^{3} y ^{4}}{1200 x ^{4} y ^{3} - 37 x}

Solution

More Steps

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

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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{2 y}{x ^{2}}

Calculate

x^{2} y^{2} 5 x^{2} y^{2} 60 = 37 x y

Simplify the expression

300 x^{4} y^{4} = 37 x y

Take the derivative of both sides

\frac{d}{d x} (300 x^{4} y^{4}) = \frac{d}{d x} (37 x y)

Calculate the derivative

More Steps

1200 x^{3} y^{4} + 1200 x^{4} y^{3} \frac{d y}{d x} = \frac{d}{d x} (37 x y)

Calculate the derivative

More Steps

1200 x^{3} y^{4} + 1200 x^{4} y^{3} \frac{d y}{d x} = 37 y + 37 x \frac{d y}{d x}

Move the expression to the left side

1200 x^{3} y^{4} + 1200 x^{4} y^{3} \frac{d y}{d x} - 37 x \frac{d y}{d x} = 37 y

Move the expression to the right side

1200 x^{4} y^{3} \frac{d y}{d x} - 37 x \frac{d y}{d x} = 37 y - 1200 x^{3} y^{4}

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

(1200 x^{4} y^{3} - 37 x) \frac{d y}{d x} = 37 y - 1200 x^{3} y^{4}

Divide both sides

\frac{( 1200 x ^{4} y ^{3} - 37 x ) \frac{d y}{d x}}{1200 x ^{4} y ^{3} - 37 x} = \frac{37 y - 1200 x ^{3} y ^{4}}{1200 x ^{4} y ^{3} - 37 x}

Divide the numbers

\frac{d y}{d x} = \frac{37 y - 1200 x ^{3} y ^{4}}{1200 x ^{4} y ^{3} - 37 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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