Solve x^2 y^2 = 2xy

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = \frac{2}{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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r = 0 r = \frac{2 sin ( 2 θ )}{∣ sin ( 2 θ ) ∣} r = - \frac{2 sin ( 2 θ )}{∣ sin ( 2 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{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

x^{2} y^{2} = 2 x y

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

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

Move the expression to the left side

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

Move the expression to the right side

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

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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\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

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

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