Solve y^2-x-2=x^2-x-2

Solve the equation

Solve for x

Solve for y

x = y x = - 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 θ = \frac{π}{4} + \frac{kπ}{2}, k \in Z

Evaluate

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

Move the expression to the left side

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

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

(sin (θ) \times r)^{2} - 2 - (cos (θ) \times r)^{2} = - 2

Factor the expression

(sin^{2} (θ) - cos^{2} (θ)) r^{2} - 2 = - 2

Simplify the expression

(2 sin^{2} (θ) - 1) r^{2} - 2 = - 2

Subtract the terms

(2 sin^{2} (θ) - 1) r^{2} - 2 - (- 2) = - 2 - (- 2)

Evaluate

(2 sin^{2} (θ) - 1) r^{2} = 0

Separate into possible cases

r^{2} = 0 2 sin^{2} (θ) - 1 = 0

Evaluate

r = 0 2 sin^{2} (θ) - 1 = 0

Solution

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r = 0 θ = \frac{π}{4} + \frac{kπ}{2}, k \in Z

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Find the derivative with respect to x

Find the derivative with respect to y

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

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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{y ^{2} - x ^{2}}{y ^{3}}

Calculate

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

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

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2 y \frac{d y}{d x} - 1 = 2 x - 1

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

2 y \frac{d y}{d x} = 2 x - 1 + 1

Since two opposites add up to 0,remove them form the expression

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

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

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{1 \times y - x \frac{d y}{d x}}{y ^{2}}

Any expression multiplied by 1 remains the same

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

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

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{y ^{2} - x ^{2}}{y ^{3}}

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