Let k be a positive integer. Consider the differential equation

\(\left\{\begin{aligned} \frac{d y}{d t} &=y^{\frac{5 k}{5 k+2}} \text { for } t>0, \\ y(0) &=0 \end{aligned}\right.\)
Which of the following statements is true?

Question

Question

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)

It has a unique solution which is continuously differentiable on (0, ∞)
It has at most two solutions which are continuously differentiable on (0, ∞).
It has infinitely many solutions which are continuously differentiable on (0, ∞).
It has no continuously differentiable solution on (0, ∞)

Answer 1

Concept:

If \(\frac{dy}{dx}=y^n\), n ∈ (0, 1) and y(a) = 0, a ∈ \(\mathbb R\) then the differential equation has infinite number of independent solution.

Explanation:

Here \( \frac{d y}{d t}=y^{\frac{5 k}{5 k+2}}\), k ∈ \(\mathbb N\), y(0) = 0

∵ n = \(\frac{5k}{5k+2}\) < 1 ∀ k ∈ \(\mathbb N\)

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

Option (3) is correct

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