Solve 12x^3y = -24

Solve the equation

Solve for x

Solve for y

x = - \frac{3 2 y ^{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 = 4 - 2 sec^{3} (θ) csc (θ) r = - 4 - 2 sec^{3} (θ) csc (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{3 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{12 y}{x ^{2}}

Calculate

12 x^{3} y = - 24

Take the derivative of both sides

\frac{d}{d x} (12 x^{3} y) = \frac{d}{d x} (- 24)

Calculate the derivative

More Steps

36 x^{2} y + 12 x^{3} \frac{d y}{d x} = \frac{d}{d x} (- 24)

Calculate the derivative

36 x^{2} y + 12 x^{3} \frac{d y}{d x} = 0

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

12 x^{3} \frac{d y}{d x} = 0 - 36 x^{2} y

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

12 x^{3} \frac{d y}{d x} = - 36 x^{2} y

Divide both sides

\frac{12 x ^{3} \frac{d y}{d x}}{12 x ^{3}} = \frac{- 36 x ^{2} y}{12 x ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{- 36 x ^{2} y}{12 x ^{3}}

Divide the numbers

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

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{3 y}{x})

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{3 \frac{d y}{d x} \times x - 3 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{3 \frac{d y}{d x} \times x - 3 y \times 1}{x ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{3 x \frac{d y}{d x} - 3 y \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

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

Use equation \frac{d y}{d x} = - \frac{3 y}{x} to substitute

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{12 y}{x ^{2}}

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