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GATE 2021 MA – Question 39

Ordinary Differential Equations · Laplace Transforms · 2 marks · Multiple choice

For y″+ay′+by=f(t),a,b>0,a≠b,a²−4b=0, with y(0)=y′(0)=0, which Laplace-transform solution holds?

  1. $y(t)=\int_0^t\tau e^{-a\tau/2}f(t-\tau)d\tau$
  2. $y(t)=\int_0^te^{-a\tau/2}f(t-\tau)d\tau$
  3. $y(t)=\int_0^t\tau e^{-b\tau/2}f(t-\tau)d\tau$
  4. $y(t)=\int_0^te^{-b\tau/2}f(t-\tau)d\tau$

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Correct answer: (A) $y(t)=\int_0^t\tau e^{-a\tau/2}f(t-\tau)d\tau$

Explanation

Zero initial conditions give Y(s)=F(s)/(s²+as+b)=F(s)/(s+a/2)². The inverse transform of the second factor is t e^(−at/2). The convolution theorem gives **A**.

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