Solve xy^2y^22x^5y-1=0

Question

Solve the equation

Solve for x

Solve for y

x = - \frac{6 32 y}{2 ∣ y ∣}, y \neq = 0 x = \frac{6 32 y}{2 ∣ y ∣}, y \neq = 0

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

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r = \frac{11 sec ^{6} ( θ ) csc ^{5} ( θ )}{11 2}

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{6 y}{5 x}

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Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{66 y}{25 x ^{2}}

Calculate

x y^{2} y^{2} 2 x^{5} y - 1 = 0

Simplify the expression

2 x^{6} y^{5} - 1 = 0

Take the derivative of both sides

\frac{d}{d x} (2 x^{6} y^{5} - 1) = \frac{d}{d x} (0)

Calculate the derivative

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12 x^{5} y^{5} + 10 x^{6} y^{4} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

12 x^{5} y^{5} + 10 x^{6} y^{4} \frac{d y}{d x} = 0

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

10 x^{6} y^{4} \frac{d y}{d x} = 0 - 12 x^{5} y^{5}

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

10 x^{6} y^{4} \frac{d y}{d x} = - 12 x^{5} y^{5}

Divide both sides

\frac{10 x ^{6} y ^{4} \frac{d y}{d x}}{10 x ^{6} y ^{4}} = \frac{- 12 x ^{5} y ^{5}}{10 x ^{6} y ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{- 12 x ^{5} y ^{5}}{10 x ^{6} y ^{4}}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{6 y}{5 x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{6 y}{5 x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{6 y}{5 x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 6 y ) \times 5 x - 6 y \times \frac{d}{d x} ( 5 x )}{( 5 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{6 \frac{d y}{d x} \times 5 x - 6 y \times \frac{d}{d x} ( 5 x )}{( 5 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{6 \frac{d y}{d x} \times 5 x - 6 y \times 5}{( 5 x ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{30 \frac{d y}{d x} \times x - 6 y \times 5}{( 5 x ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{30 \frac{d y}{d x} \times x - 30 y}{( 5 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{30 x \frac{d y}{d x} - 30 y}{( 5 x ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{30 x \frac{d y}{d x} - 30 y}{25 x ^{2}}

Calculate

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

Use equation \frac{d y}{d x} = - \frac{6 y}{5 x} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{6 x ( - \frac{6 y}{5 x} ) - 6 y}{5 x ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{66 y}{25 x ^{2}}

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