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GATE 2023 IN – Question 20

Engineering Mathematics · Differential equations · 1 mark · Multiple choice

The solution of $\ddot x=-k\dot x$, $k>0$, with $x(0)=1$, $\dot x(0)=0$, for $t\ge0$, is

  1. $x(t)=2e^{-kt}+2kt-1$
  2. $x(t)=2e^{-kt}-1$
  3. $x(t)=1$
  4. $x(t)=2e^{-kt}-kt-1$

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Correct answer: (C) $x(t)=1$

Explanation

Let $v=\dot x$. Its equation is $\dot v=-kv$, giving $v(t)=v(0)e^{-kt}=0$. Integrating and using $x(0)=1$ yields $x(t)=1$, **C**.