Question

Let \(f\) be a differential function with
\[ \lim_{x \to \infty} f(x) = 0. \text{ If } y' + y f'(x) - f(x) f'(x) = 0, \lim_{x \to \infty} y(x) = 0 \text{ then,} \]

• A. \(y + 1 = e^{-f(x)} + f(x)\)
• B. \(y + 1 = e^{-f(x)} + f(x)\)
• C. \(y + 2 = e^{-f(x)} + f(x)\)
• D. \(y - 1 = e^{-f(x)} + f(x)\)

Hint:

To solve such differential equations, look for terms that approach zero as x increases. The limits often simplify the function form.