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GATE 2015 EC – Question 36

Engineering Mathematics · Differential Equations · 2 marks · Multiple choice

The solution of the differential equation $\frac{d^2y}{dt^2} + 2\frac{dy}{dt} + y = 0$ with $y(0) = y'(0) = 1$ is

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

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Show answer and explanation

Correct answer: (B) $(1 + 2t)e^{-t}$

Explanation

The characteristic equation $r^2 + 2r + 1 = 0$ has a repeated root $-1$, so $y = (A + Bt)e^{-t}$. From $y(0) = 1$ we get $A = 1$. Then $y'(0) = -A + B = 1$ gives $B = 2$, so $y = (1 + 2t)e^{-t}$.