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.

Answer 1

The Correct Option is B

1. Step 1: Start with the given differential equation:

\[ y' + y f'(x) - f(x) f'(x) = 0 \]

1. This is a first-order linear differential equation involving both \( y \) and \( f(x) \).
2. Step 2: Rearrange the terms to isolate \( y' \):

\[ y' = f'(x) \left( f(x) - y \right) \]

1. Now, we have an equation that describes how \( y \) changes with respect to \( x \), in terms of the function \( f(x) \) and its derivative.
2. Step 3: Notice that the form of the equation suggests that \( y \) might be related to \( f(x) \) in some way. Since \( f(x) \to 0 \) as \( x \to \infty \), we expect \( y(x) \) to simplify as well, possibly to a constant.
3. Step 4: Assume that as \( x \to \infty \), the behavior of the functions becomes simpler. Specifically, since \( \lim_{x \to \infty} f(x) = 0 \), we hypothesize that \( y(x) \) may behave similarly to an exponential function that decays as \( f(x) \) does.
4. Step 5: Assume a solution for \( y(x) \) in the form:

\[ y + 1 = e^{-f(x)} + f(x) \]

1. This form satisfies the differential equation and the condition that as \( x \to \infty \), \( y(x) \to 0 \) because \( f(x) \to 0 \).
2. Step 6: Substitute this proposed solution into the original differential equation to verify that it satisfies the equation. If both sides of the equation balance, then our assumption is correct.

\[ y' + y f'(x) - f(x) f'(x) = 0 \]

1. After substituting \( y = e^{-f(x)} + f(x) - 1 \) into this equation, we find that it holds true, confirming the correctness of our solution.
2. Step 7: Therefore, the correct relationship is:

\[ y + 1 = e^{-f(x)} + f(x) \]