Solve 6x y=4x 11y

Question

Solve the equation

Solve for x

Solve for y

x = 0

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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 = \frac{kπ}{2}, 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{y}{x}

Calculate

6 x y = 4 x 11 y

Simplify the expression

6 x y = 44 x y

Take the derivative of both sides

\frac{d}{d x} (6 x y) = \frac{d}{d x} (44 x y)

Calculate the derivative

More Steps

6 y + 6 x \frac{d y}{d x} = \frac{d}{d x} (44 x y)

Calculate the derivative

More Steps

6 y + 6 x \frac{d y}{d x} = 44 y + 44 x \frac{d y}{d x}

Move the expression to the left side

6 y + 6 x \frac{d y}{d x} - 44 x \frac{d y}{d x} = 44 y

Move the expression to the right side

6 x \frac{d y}{d x} - 44 x \frac{d y}{d x} = 44 y - 6 y

Add and subtract

More Steps

- 38 x \frac{d y}{d x} = 44 y - 6 y

Add and subtract

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- 38 x \frac{d y}{d x} = 38 y

Divide both sides

\frac{- 38 x \frac{d y}{d x}}{- 38 x} = \frac{38 y}{- 38 x}

Divide the numbers

\frac{d y}{d x} = \frac{38 y}{- 38 x}

Solution

More Steps

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

Calculate

6 x y = 4 x 11 y

Simplify the expression

6 x y = 44 x y

Take the derivative of both sides

\frac{d}{d x} (6 x y) = \frac{d}{d x} (44 x y)

Calculate the derivative

More Steps

6 y + 6 x \frac{d y}{d x} = \frac{d}{d x} (44 x y)

Calculate the derivative

More Steps

6 y + 6 x \frac{d y}{d x} = 44 y + 44 x \frac{d y}{d x}

Move the expression to the left side

6 y + 6 x \frac{d y}{d x} - 44 x \frac{d y}{d x} = 44 y

Move the expression to the right side

6 x \frac{d y}{d x} - 44 x \frac{d y}{d x} = 44 y - 6 y

Add and subtract

More Steps

- 38 x \frac{d y}{d x} = 44 y - 6 y

Add and subtract

More Steps

- 38 x \frac{d y}{d x} = 38 y

Divide both sides

\frac{- 38 x \frac{d y}{d x}}{- 38 x} = \frac{38 y}{- 38 x}

Divide the numbers

\frac{d y}{d x} = \frac{38 y}{- 38 x}

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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