Solve 2x^2 4x=19-3y^2

Question

Solve the equation

Solve for x

Solve for y

x = \frac{3 19 - 3 y ^{2}}{2}

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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 derivative with respect to x

Find the derivative with respect to y

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

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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{8 x y ^{2} + 16 x ^{4}}{y ^{3}}

Calculate

2 x^{2} 4 x = 19 - 3 y^{2}

Simplify the expression

8 x^{3} = 19 - 3 y^{2}

Take the derivative of both sides

\frac{d}{d x} (8 x^{3}) = \frac{d}{d x} (19 - 3 y^{2})

Calculate the derivative

More Steps

24 x^{2} = \frac{d}{d x} (19 - 3 y^{2})

Calculate the derivative

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

Swap the sides of the equation

- 6 y \frac{d y}{d x} = 24 x^{2}

Divide both sides

\frac{- 6 y \frac{d y}{d x}}{- 6 y} = \frac{24 x ^{2}}{- 6 y}

Divide the numbers

\frac{d y}{d x} = \frac{24 x ^{2}}{- 6 y}

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{8 x y - 4 x ^{2} \times \frac{d}{d x} ( y )}{y ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{8 x y - 4 x ^{2} \frac{d y}{d x}}{y ^{2}}

Use equation \frac{d y}{d x} = - \frac{4 x ^{2}}{y} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{8 x y - 4 x ^{2} ( - \frac{4 x ^{2}}{y} )}{y ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{8 x y ^{2} + 16 x ^{4}}{y ^{3}}

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