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GATE 2024 EC – Question 11

Engineering Mathematics · Differential Equations · 1 mark · Multiple choice

The general form of the complementary function of a differential equation is given by $y(t)=(At+B)e^{-2t}$, where $A$ and $B$ are real constants determined by the initial condition. The corresponding differential equation is ______.

  1. $\dfrac{d^2y}{dt^2}+4\dfrac{dy}{dt}+4y=f(t)$
  2. $\dfrac{d^2y}{dt^2}+4y=f(t)$
  3. $\dfrac{d^2y}{dt^2}+3\dfrac{dy}{dt}+2y=f(t)$
  4. $\dfrac{d^2y}{dt^2}+5\dfrac{dy}{dt}+6y=f(t)$

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Correct answer: (A) $\dfrac{d^2y}{dt^2}+4\dfrac{dy}{dt}+4y=f(t)$

Explanation

$(At+B)e^{-2t}$ is the form for a repeated characteristic root $-2$, i.e. characteristic polynomial $(s+2)^2=s^2+4s+4$. The differential equation is therefore $y''+4y'+4y=f(t)$.