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GATE 2025 BT – Question 45

Engineering Mathematics · Differential Equations: First order ODEs, higher order ODEs with constant coefficients, Cauchy and Euler equations, Laplace transforms · 2 marks · Multiple choice

Let $m$ and $n$ be fixed real numbers. If the function $y(t) = C_1e^t + C_2e^{-t}$ is a solution of $\frac{d^2y}{dt^2} + m\frac{dy}{dt} + ny = 0$ for any constants $C_1$ and $C_2$, then $m + n$ is equal to

  1. -2
  2. -1
  3. 0
  4. 1

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Correct answer: (B) -1

Explanation

The solution has the characteristic roots $+1$ and $-1$, so the characteristic equation is $r^2 - 1 = 0$. Comparing with $r^2 + mr + n = 0$ gives $m = 0$ and $n = -1$, so $m + n = -1$.