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?
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?
2. Every solution of the IVP is bounded on its maximal interval of existence.
Which one of the following is correct?
Show Hint
Check Lipschitz continuity for uniqueness and analyze growth for boundedness in IVPs.
Updated On: Feb 1, 2025
- Both I and II are TRUE
- I is TRUE and II is FALSE
- I is FALSE and II is TRUE
- Both I and II are FALSE
Show Solution
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The Correct Option is B
Solution and Explanation
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 \( {(2)} \).
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