Question

Let \( y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} = (y - 1)(y - 3) \] satisfying the condition \( y(0) = 2 \). Then which of the following is/are TRUE?

• A. The function \( y(x) \) is not bounded above.
• B. The function \( y(x) \) is bounded.
• C. \( \lim_{x \to \infty} y(x) = 1 \)
• D. \( \lim_{x \to \infty} y(x) = 3 \)

Hint:

For separable differential equations, first separate the variables, integrate, and then apply initial conditions to find the general solution.

Answer 1

The Correct Option is B, C, D

Step 1: Solving the differential equation.
The differential equation is separable. We separate the variables as follows: \[ \frac{dy}{(y - 1)(y - 3)} = dx. \] We perform partial fraction decomposition to solve the left-hand side and integrate: \[ \frac{1}{(y - 1)(y - 3)} = \frac{A}{y - 1} + \frac{B}{y - 3}. \] Solving for \( A \) and \( B \), we get: \[ \frac{1}{(y - 1)(y - 3)} = \frac{1}{2(y - 1)} - \frac{1}{2(y - 3)}. \] Integrating both sides, we get: \[ \frac{1}{2} \ln \left| \frac{y - 1}{y - 3} \right| = x + C. \]
Step 2: Applying the initial condition.
Using \( y(0) = 2 \), we can solve for \( C \): \[ \frac{1}{2} \ln \left| \frac{2 - 1}{2 - 3} \right| = 0 + C, \] which gives \( C = \ln 1 = 0 \).
Step 3: Analyzing the long-term behavior.
As \( x \to \infty \), \( y(x) \) approaches 3 because the solution tends to the stable equilibrium point at \( y = 3 \).
Step 4: Conclusion.
Thus, the correct answers are (B), (C), and (D).