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GATE 2025 BT – Question 13

Engineering Mathematics · Differential Equations: First order ODEs, higher order ODEs with constant coefficients, Cauchy and Euler equations, Laplace transforms · 1 mark · Multiple choice

Let $y(t)$ be a bacterial population whose growth is given by $\frac{dy}{dt} = \lambda(y + 2)$ where $\lambda$ is the growth rate constant. If $y(0) = 1$ and $y(1) = 4$, then the value of $\lambda$ is

  1. ln 2
  2. ln 3
  3. ln 4
  4. ln 6

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Correct answer: (A) ln 2

Explanation

Separating the variables gives $y + 2 = (y_0 + 2)e^{\lambda t} = 3e^{\lambda t}$. At $t = 1$, $y + 2 = 6$, so $e^{\lambda} = 2$ and $\lambda = \ln 2$.