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Let f ∶ ℝ2 → ℝ be a locally Lipschitz function. Consider the initial value problem

ẋ = f(t, x), x(t0) = x0

for (t0, x0) ∈ ℝ2. Suppose that J(t0, x0) represents the maximal interval of existence for the initial value problem. Which of the following statements is true?

This question was previously asked in
CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. J(t0, x0) = ℝ.
  2. J(t0, x0) is an open set.
  3. J(t0, x0) is a closed set.
  4. J(t0, x0) could be an empty set.

Answer (Detailed Solution Below)

Option 2 : J(t0, x0) is an open set.
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Detailed Solution

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Explanation:

Let f ∶ ℝ2 → ℝ be a locally Lipschitz function. Consider the initial value problem

ẋ = f(t, x), x(t0) = x0

for (t0, x0) ∈ ℝ2. 

By using Picard's theorem we know that solutions lie in the interval 

\(|x-x_o|< a ; |t-t_o|< b \)

which is an open interval 

Therefore, the Correct Option is Option (2).

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Last updated on Jul 8, 2025

-> The CSIR NET June 2025 Exam Schedule has been released on its official website.The exam will be held on 28th July 2025.

-> The CSIR UGC NET is conducted in five subjects -Chemical Sciences, Earth Sciences, Life Sciences, Mathematical Sciences, and Physical Sciences. 

-> Postgraduates in the relevant streams can apply for this exam.

-> Candidates must download and practice questions from the CSIR NET Previous year papers. Attempting the CSIR NET mock tests are also very helpful in preparation.

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