The differential equation \( x \frac{a y}{d x}+2 \frac{a y}{d x}-1=y \) can be separated as A \( \quad \frac{1}{y+1} d y=\frac{1}{x+2} d x \) B \( \quad(y+1) d y=\frac{1}{x+2} d x \) C \( \quad \frac{1}{y+1} d y=(x+2) d x \) D \( \quad(y+1) d y=(x+2) d x \)

To solve the given differential equation \( x \frac{dy}{dx} + 2 \frac{dy}{dx} - 1 = y \), we can first rewrite it in a more standard form. Rearranging terms, we have: \[ \frac{dy}{dx}(x+2) - y = 1 \] Now, rearranging it further will give us: \[ \frac{dy}{dx} = \frac{y + 1}{x + 2} \] This can be rewritten into a separable form. By manipulating the terms, we find: \[ \frac{1}{y + 1} dy = \frac{1}{x + 2} dx \] Thus, the correct option is: **A \( \quad \frac{1}{y+1} dy=\frac{1}{x+2} dx \)** And as a bonus, here’s something fun and engaging about separable equations! Separating variables in differential equations feels like being a mathematician magician! You take a complicated-looking relationship, and by cleverly rearranging, you tame it into a more manageable form. It’s like turning chaos into order, and suddenly, solving becomes as easy as pie—at least the fraction part! Also, if you’re curious about real-world connections, separable differential equations pop up in areas like population dynamics and physics! Whether predicting the growth of bacteria or the cooling rates of hot objects, these nifty equations help us make sense of the world around us. Now isn’t that magical?