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Let k be a positive integer. Consider the differential equation


Which of the following statements is true?

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. It has a unique solution which is continuously differentiable on (0, ∞)
  2. It has at most two solutions which are continuously differentiable on (0, ∞).

  3. It has infinitely many solutions which are continuously differentiable on (0, ∞).

  4. It has no continuously differentiable solution on (0, ∞)

Answer (Detailed Solution Below)

Option 3 :

It has infinitely many solutions which are continuously differentiable on (0, ∞).
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Detailed Solution

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Concept:

If , n ∈ (0, 1) and y(a) = 0, a ∈  then the differential equation has infinite number of independent solution.

Explanation:

Here , k ∈ , y(0) = 0

∵ n =   < 1 ∀ k ∈ 

Hence It has infinitely many solutions which are continuously differentiable on (0, ∞).

Option (3) is correct

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