Solve xy xy^4 - 2 = -1

Question

Solve the equation

y = \frac{5 x ^{3}}{x}

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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{1}{7 cos ^{2} ( θ ) sin ^{5} ( θ )}

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{2 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{14 y}{25 x ^{2}}

Calculate

x y x y^{4} - 2 = - 1

Simplify the expression

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

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

2 x y^{5} + 5 x^{2} y^{4} \frac{d y}{d x} = 0

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

5 x^{2} y^{4} \frac{d y}{d x} = 0 - 2 x y^{5}

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

5 x^{2} y^{4} \frac{d y}{d x} = - 2 x y^{5}

Divide both sides

\frac{5 x ^{2} y ^{4} \frac{d y}{d x}}{5 x ^{2} y ^{4}} = \frac{- 2 x y ^{5}}{5 x ^{2} y ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x y ^{5}}{5 x ^{2} y ^{4}}

Divide the numbers

More Steps

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{10 \frac{d y}{d x} \times x - 2 y \times 5}{( 5 x ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{10 \frac{d y}{d x} \times x - 10 y}{( 5 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{10 x \frac{d y}{d x} - 10 y}{( 5 x ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{10 x \frac{d y}{d x} - 10 y}{25 x ^{2}}

Calculate

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

Use equation \frac{d y}{d x} = - \frac{2 y}{5 x} to substitute

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{14 y}{25 x ^{2}}

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