Solve 2y-8xy=0 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = \frac{1}{4}

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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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r = 0 r = \frac{sec ( θ )}{4}

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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 y}{1 - 4 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{32 y}{1 - 8 x + 16 x ^{2}}

Calculate

2 y - 8 x y = 0

Take the derivative of both sides

\frac{d}{d x} (2 y - 8 x y) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

2 \frac{d y}{d x} - 8 y - 8 x \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 \frac{d y}{d x} - 8 y - 8 x \frac{d y}{d x} = 0

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

(2 - 8 x) \frac{d y}{d x} - 8 y = 0

Move the constant to the right side

(2 - 8 x) \frac{d y}{d x} = 0 + 8 y

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

More Steps

\frac{d y}{d x} = \frac{4 y}{1 - 4 x}

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

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

Calculate

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

Calculate

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

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

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{32 y}{1 - 8 x + 16 x ^{2}}

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