Solve |x|^5/2 |y|^5/2 = 1

Question

Solve the equation

Solve for x

Solve for y

x = \frac{5 4}{y} x = - \frac{5 4}{y}

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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 derivative with respect to x

Find the derivative with respect to y

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

Calculate

\frac{∣ x ∣ ^{5}}{2} \frac{∣ y ∣ ^{5}}{2} = 1

Simplify the expression

\frac{x ^{5} y ^{5}}{4} = 1

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

\frac{x ^{5} y ^{5} ( 5 x ^{4} y ^{5} + 5 x ^{5} y ^{4} \frac{d y}{d x} )}{4 ∣ x ^{5} y ^{5} ∣} = 0

Simplify

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

Rewrite the expression

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

Move the constant to the right side

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

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

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

Divide both sides

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

Divide the numbers

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

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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