Solve 2y-3=9-4x x y

Question

Solve the equation

Solve for x

Solve for y

x = \frac{- 2 y ^{2} + 12 y}{2 ∣ y ∣} x = - \frac{- 2 y ^{2} + 12 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

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 y}{1 + 2 x ^{2}}

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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{4 y - 24 y x ^{2}}{1 + 4 x ^{2} + 4 x ^{4}}

Calculate

2 y - 3 = 9 - 4 xx y

Simplify the expression

2 y - 3 = 9 - 4 x^{2} y

Take the derivative of both sides

\frac{d}{d x} (2 y - 3) = \frac{d}{d x} (9 - 4 x^{2} y)

Calculate the derivative

More Steps

2 \frac{d y}{d x} = \frac{d}{d x} (9 - 4 x^{2} y)

Calculate the derivative

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

Move the variable to the left side

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

Collect like terms by calculating the sum or difference of their coefficients

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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\frac{d y}{d x} = - \frac{4 x y}{1 + 2 x ^{2}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{4 x y}{1 + 2 x ^{2}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{4 x y}{1 + 2 x ^{2}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 4 x y ) \times ( 1 + 2 x ^{2} ) - 4 x y \times \frac{d}{d x} ( 1 + 2 x ^{2} )}{( 1 + 2 x ^{2} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 4 y + 4 x \frac{d y}{d x} ) ( 1 + 2 x ^{2} ) - 4 x y \times \frac{d}{d x} ( 1 + 2 x ^{2} )}{( 1 + 2 x ^{2} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 4 y + 4 x \frac{d y}{d x} ) ( 1 + 2 x ^{2} ) - 4 x y \times 4 x}{( 1 + 2 x ^{2} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y + 8 y x ^{2} + 4 x \frac{d y}{d x} + 8 x ^{3} \frac{d y}{d x} - 4 x y \times 4 x}{( 1 + 2 x ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y + 8 y x ^{2} + 4 x \frac{d y}{d x} + 8 x ^{3} \frac{d y}{d x} - 16 x ^{2} y}{( 1 + 2 x ^{2} ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - 8 y x ^{2} + 4 x \frac{d y}{d x} + 8 x ^{3} \frac{d y}{d x}}{( 1 + 2 x ^{2} ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{4 x y}{1 + 2 x ^{2}} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - 8 y x ^{2} + 4 x ( - \frac{4 x y}{1 + 2 x ^{2}} ) + 8 x ^{3} ( - \frac{4 x y}{1 + 2 x ^{2}} )}{( 1 + 2 x ^{2} ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - 24 y x ^{2}}{1 + 4 x ^{2} + 4 x ^{4}}

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