Solve y^3-3x^2+5=0

Question

Solve the equation

$Solve for x$

$Solve for y$

x = \frac{3 y ^{3} + 15}{3} x = - \frac{3 y ^{3} + 15}{3}

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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

Not symmetry with respect to the origin

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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{2 x}{y ^{2}}

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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 ^{3} - 8 x ^{2}}{y ^{5}}

Calculate

y^{3} - 3 x^{2} + 5 = 0

Take the derivative of both sides

\frac{d}{d x} (y^{3} - 3 x^{2} + 5) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

3 y^{2} \frac{d y}{d x} - 6 x = \frac{d}{d x} (0)

Calculate the derivative

3 y^{2} \frac{d y}{d x} - 6 x = 0

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

3 y^{2} \frac{d y}{d x} = 0 + 6 x

Add the terms

3 y^{2} \frac{d y}{d x} = 6 x

Divide both sides

\frac{3 y ^{2} \frac{d y}{d x}}{3 y ^{2}} = \frac{6 x}{3 y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{6 x}{3 y ^{2}}

Cancel out the common factor 3

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{2 y ^{2} - 4 x y \frac{d y}{d x}}{( y ^{2} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{2 y ^{2} - 4 x y \frac{d y}{d x}}{y ^{4}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{2 y - 4 x \frac{d y}{d x}}{y ^{3}}

Use equation \frac{d y}{d x} = \frac{2 x}{y ^{2}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{2 y - 4 x \times \frac{2 x}{y ^{2}}}{y ^{3}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{2 y ^{3} - 8 x ^{2}}{y ^{5}}

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Y^3 3x^2+5=0

Solve the equation

Solve for x

Solve for y

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

$Solve for x$

$Solve for y$

$Find the derivative with respect to x$

$Find the derivative with respect to y$

$Find the second derivative with respect to x$

$Find the second derivative with respect to y$