Question

Let \( y_c : \mathbb{R} \to (0, \infty) \) be the solution of the Bernoulli’s equation \[ \frac{dy}{dx} - y + y^3 = 0, \quad y(0) = c>0. \] Then, for every \( c>0 \), which one of the following is true?

• A. \( \lim_{x \to \infty} y_c(x) = 0
• B. \( \lim_{x \to \infty} y_c(x) = 1
• C. \( \lim_{x \to \infty} y_c(x) = e
• D. \( \lim_{x \to \infty} y_c(x) \text{ does not exist}

Hint:

In Bernoulli’s equation, the solution typically stabilizes at a fixed value, which can be found by setting the derivative equal to zero and solving for \( y \).

Answer 1

The Correct Option is B

The given Bernoulli’s equation can be written as: \[ \frac{dy}{dx} = y - y^3. \] As \( x \to \infty \), the solution \( y_c(x) \) tends to a steady-state value where \( \frac{dy}{dx} = 0 \). Solving for this steady state, we get: \[ y - y^3 = 0 \quad \Rightarrow \quad y(1 - y^2) = 0. \] Since \( y>0 \), we find that \( y = 1 \). Therefore, as \( x \to \infty \), \( y_c(x) \to 1 \).