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GATE 2017 CE (CE1) – Question 37

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 2 marks · Multiple choice

The solution of $dQ/dt+Q=1$ with $Q(0)=0$ is

  1. $Q=e^{-t}-1$
  2. $Q=1+e^{-t}$
  3. $Q=1-e^t$
  4. $Q=1-e^{-t}$

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Correct answer: (D) $Q=1-e^{-t}$

Explanation

The integrating factor is $e^t$, giving $d(e^tQ)/dt=e^t$. Integration yields $Q=1+Ce^{-t}$. The initial condition gives $0=1+C$, so $C=-1$. Thus $Q(t)=1-e^{-t}$, D. Its derivative $e^{-t}$ plus itself equals 1, providing a direct check.

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