Question

Separating variable in the equation gives you the following equation:​

Answer 1

Answer:

$\textsf{E.} \quad y\:dy=xe^x\:dx$

Step-by-step explanation:

Given equation:

$ye^{-x}\dfrac{dy}{dx}=x$

Divide both sides by $e^{-x}$:

$\implies \dfrac{ye^{-x}}{e^{-x}}\:\dfrac{dy}{dx}=\dfrac{x}{e^{-x}}$

$\implies y\:\dfrac{dy}{dx}=\dfrac{x}{e^{-x}}$

Multiply both sides by $dx$ :

$\implies y\:\dfrac{dy}{dx}\cdot dx=\dfrac{x}{e^{-x}}\cdot dx$

$\implies y\:dy=\dfrac{x}{e^{-x}}\:dx$

$\textsf{Apply exponent rule} \quad \dfrac{1}{a^{-n}}=a^n:$

$\implies y\:dy=xe^x\:dx$

Answer 2

Answer: E

Step-by-step explanation:

$ye^{-x} \frac{dy}{dx}=x\\\\y \frac{dy}{dx}=xe^{x} \\ \\ \boxed{y dy=xe^{x} dx}$