Discussion

(y-x)∂x+4xy∂y=0 is a

Answer: B

Linearity is judged in the dependent variable and its derivatives: they may appear only to the first power and never multiplied together.

\[ (y-x)\,\partial x + 4xy\,\partial y = 0 \]

Rearranged as

\[ \frac{dy}{dx} = \frac{x-y}{4xy} \]

the term \( 4xy\,\tfrac{dy}{dx} \) has \( y \) multiplying its own derivative, so the equation is non-linear.

A linear first-order equation has the form \( \tfrac{dy}{dx} + P(x)y = Q(x) \), with no product of \( y \) and \( y' \).

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