etermine whether the given differential equation is separable.

left parenthesis xy Superscript $4$ Baseline plus $6$ y Superscript $4$ Baseline right parenthesis dy minus $2$ ydx equals $0$

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Part $1$

Is the differential equation$$ separable?

A$.$

$$Yes; because

StartFraction dy Over dx EndFraction equals g left parenthesis x right parenthesis p left parenthesis y right parenthesis

where

$$g$($x$)$equalsStartFraction $2$ y Over x plus $6$ EndFraction

and

$$p$($y$)$equalsStartFraction $1$ Over y Superscript $4$ EndFraction.

B$.$

$$Yes; because

StartFraction dy Over dx EndFraction equals g left parenthesis x right parenthesis p left parenthesis y right parenthesis

where

$$g$($x$)$equalsStartFraction $1$ Over x plus $6$ EndFraction

and

$$p$($y$)$equalsStartFraction $2$ Over y cubed EndFraction.

C$.$

$$Yes; because

StartFraction dy Over dx EndFraction equals g left parenthesis x right parenthesis p left parenthesis y right parenthesis

where

$$g$($x$)$equalsStartFraction $1$ Over x plus $6$ EndFraction

and

$$p$($y$)$equalsStartFraction $1$ Over y cubed EndFraction.