Question

Consider the initial value problem (IVP): \[ \frac{dy}{dx} = e^{-y}, \quad y(0) = 0. \] 1. The IVP has a unique solution on \( {R} \).
2. Every solution of the IVP is bounded on its maximal interval of existence.
Which one of the following is correct?

• A. Both I and II are TRUE
• B. I is TRUE and II is FALSE
• C. I is FALSE and II is TRUE
• D. Both I and II are FALSE

Hint:

Check Lipschitz continuity for uniqueness and analyze growth for boundedness in IVPs.

Answer 1

The Correct Option is B

Step 1: Uniqueness of the solution. The differential equation satisfies the Lipschitz condition, ensuring that the IVP has a unique solution on \( {R} \). Step 2: Boundedness. As \( x \to \infty \), the solution \( y(x) \) becomes unbounded due to the growth properties of the equation. Step 3: Conclusion. Statement I is true, but statement II is false. The correct answer is \( {(2)} \).