Solve 31x^24=18xy

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = \frac{9 y}{62}

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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 = {arctan (\frac{62}{9}) + kπ \frac{π}{2} + kπ, k \in Z

Evaluate

31 x^{2} \times 4 = 18 x y

Evaluate

124 x^{2} = 18 x y

Move the expression to the left side

124 x^{2} - 18 x y = 0

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

124 (cos (θ) \times r)^{2} - 18 cos (θ) \times r sin (θ) \times r = 0

Factor the expression

(124 cos^{2} (θ) - 18 cos (θ) sin (θ)) r^{2} = 0

Simplify the expression

(124 cos^{2} (θ) - 9 sin (2 θ)) r^{2} = 0

Separate into possible cases

r^{2} = 0 124 cos^{2} (θ) - 9 sin (2 θ) = 0

Evaluate

r = 0 124 cos^{2} (θ) - 9 sin (2 θ) = 0

Solution

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r = 0 θ = {arctan (\frac{62}{9}) + kπ \frac{π}{2} + 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{124 x - 9 y}{9 x}

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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{- 124 x + 18 y}{9 x ^{2}}

Calculate

31 x^{2} 4 = 18 x y

Simplify the expression

124 x^{2} = 18 x y

Take the derivative of both sides

\frac{d}{d x} (124 x^{2}) = \frac{d}{d x} (18 x y)

Calculate the derivative

More Steps

248 x = \frac{d}{d x} (18 x y)

Calculate the derivative

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248 x = 18 y + 18 x \frac{d y}{d x}

Swap the sides of the equation

18 y + 18 x \frac{d y}{d x} = 248 x

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

18 x \frac{d y}{d x} = 248 x - 18 y

Divide both sides

\frac{18 x \frac{d y}{d x}}{18 x} = \frac{248 x - 18 y}{18 x}

Divide the numbers

\frac{d y}{d x} = \frac{248 x - 18 y}{18 x}

Divide the numbers

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\frac{d y}{d x} = \frac{124 x - 9 y}{9 x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{124 x - 9 y}{9 x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{124 x - 9 y}{9 x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 124 x - 9 y ) \times 9 x - ( 124 x - 9 y ) \times \frac{d}{d x} ( 9 x )}{( 9 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{( 124 - 9 \frac{d y}{d x} ) \times 9 x - ( 124 x - 9 y ) \times \frac{d}{d x} ( 9 x )}{( 9 x ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 124 - 9 \frac{d y}{d x} ) \times 9 x - ( 124 x - 9 y ) \times 9}{( 9 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{1116 x - 81 x \frac{d y}{d x} - ( 124 x - 9 y ) \times 9}{( 9 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{1116 x - 81 x \frac{d y}{d x} - ( 1116 x - 81 y )}{( 9 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 81 x \frac{d y}{d x} + 81 y}{( 9 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 81 x \frac{d y}{d x} + 81 y}{81 x ^{2}}

Calculate

\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{124 x - 9 y}{9 x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- x \times \frac{124 x - 9 y}{9 x} + y}{x ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 124 x + 18 y}{9 x ^{2}}

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