Solve x^2*2009=y^2

Solve the equation

Solve for x

Solve for y

x = \frac{41 \times ∣ y ∣}{287} x = - \frac{41 \times ∣ y ∣}{287}

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Testing for symmetry

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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Rewrite the equation

r = 0 θ = ⎩ ⎨ ⎧ - arccos (- \frac{2010}{2010}) + kπ - arccos (\frac{2010}{2010}) + kπ, k \in Z

Evaluate

x^{2} \times 2009 = y^{2}

Use the commutative property to reorder the terms

2009 x^{2} = y^{2}

Move the expression to the left side

2009 x^{2} - y^{2} = 0

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

2009 (cos (θ) \times r)^{2} - (sin (θ) \times r)^{2} = 0

Factor the expression

(2009 cos^{2} (θ) - sin^{2} (θ)) r^{2} = 0

Simplify the expression

(2010 cos^{2} (θ) - 1) r^{2} = 0

Separate into possible cases

r^{2} = 0 2010 cos^{2} (θ) - 1 = 0

Evaluate

r = 0 2010 cos^{2} (θ) - 1 = 0

Solution

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Evaluate

2010 cos^{2} (θ) - 1 = 0

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

2010 cos^{2} (θ) = 0 + 1

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

2010 cos^{2} (θ) = 1

Divide both sides

\frac{2010 cos ^{2} ( θ )}{2010} = \frac{1}{2010}

Divide the numbers

cos^{2} (θ) = \frac{1}{2010}

Take the root of both sides of the equation and remember to use both positive and negative roots

cos (θ) = \pm \frac{1}{2010}

Simplify the expression

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cos (θ) = \pm \frac{2010}{2010}

Separate the equation into 2 possible cases

cos (θ) = \frac{2010}{2010} cos (θ) = - \frac{2010}{2010}

Calculate

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θ = ⎩ ⎨ ⎧ - arccos (\frac{2010}{2010}) + 2 kπ arccos (\frac{2010}{2010}) + 2 kπ, k \in Z cos (θ) = - \frac{2010}{2010}

Calculate

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θ = ⎩ ⎨ ⎧ - arccos (\frac{2010}{2010}) + 2 kπ arccos (\frac{2010}{2010}) + 2 kπ, k \in Z θ = ⎩ ⎨ ⎧ - arccos (- \frac{2010}{2010}) + 2 kπ arccos (- \frac{2010}{2010}) + 2 kπ, k \in Z

Find the union

θ = ⎩ ⎨ ⎧ - arccos (- \frac{2010}{2010}) + kπ - arccos (\frac{2010}{2010}) + kπ, k \in Z

r = 0 θ = ⎩ ⎨ ⎧ - arccos (- \frac{2010}{2010}) + kπ - arccos (\frac{2010}{2010}) + kπ, k \in Z

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{2009 x}{y}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{2009 y ^{2} - 4036081 x ^{2}}{y ^{3}}

Calculate

x^{2} \cdot 2009 = y^{2}

Simplify the expression

2009 x^{2} = y^{2}

Take the derivative of both sides

\frac{d}{d x} (2009 x^{2}) = \frac{d}{d x} (y^{2})

Calculate the derivative

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4018 x = \frac{d}{d x} (y^{2})

Calculate the derivative

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4018 x = 2 y \frac{d y}{d x}

Swap the sides of the equation

2 y \frac{d y}{d x} = 4018 x

Divide both sides

\frac{2 y \frac{d y}{d x}}{2 y} = \frac{4018 x}{2 y}

Divide the numbers

\frac{d y}{d x} = \frac{4018 x}{2 y}

Cancel out the common factor 2

\frac{d y}{d x} = \frac{2009 x}{y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{2009 x}{y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{2009 x}{y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 2009 x ) \times y - 2009 x \times \frac{d}{d x} ( y )}{y ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{2009 y - 2009 x \times \frac{d}{d x} ( y )}{y ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{2009 y - 2009 x \frac{d y}{d x}}{y ^{2}}

Use equation \frac{d y}{d x} = \frac{2009 x}{y} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{2009 y - 2009 x \times \frac{2009 x}{y}}{y ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{2009 y ^{2} - 4036081 x ^{2}}{y ^{3}}

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