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GATE 2020 MA – Question 13

Ordinary Differential Equations · First Order Equations and Initial Value Problems · 1 mark · Multiple choice

The IVP is $y\prime=y^{3/5}$, y(0) = b. It has

  1. A unique solution if b = 0.
  2. No solution if b = 1.
  3. Infinitely many solutions if b = 2.
  4. A unique solution if b = 1.

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Correct answer: (D) A unique solution if b = 1.

Explanation

The right side is locally Lipschitz in y away from zero, so the IVP with b = 1 has a unique local solution. At b = 0, both the zero solution and delayed nonzero branches exist because Lipschitz continuity fails there. Nonzero initial values do not give the nonuniqueness asserted in C. Thus D under the standard local-IVP interpretation.

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