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GATE 2025 MA – Question 12

Ordinary Differential Equations · Laplace Transforms · 1 mark · Multiple choice

Let Y(s) be the Laplace transform of Bessel J₀(x), with transforms of J₀ and its first two derivatives assumed to exist. Which relation holds for s>0?

  1. $Y^\prime+2sY/(s^2+1)=0$
  2. $Y^\prime-2sY/(s^2+1)=0$
  3. $Y^\prime-sY/(s^2+1)=0$
  4. $Y^\prime+sY/(s^2+1)=0$

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Correct answer: (D) $Y^\prime+sY/(s^2+1)=0$

Explanation

Use $xJ_0^{\prime\prime}+J_0^\prime+xJ_0=0$, J₀(0)=1,J₀′(0)=0, and L{xf}=−dL{f}/ds. This gives $-(s^2Y-s)^\prime+(sY-1)-Y^\prime=0$, hence $(s^2+1)Y^\prime+sY=0$. **D**.

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