GATE 2023 ME – Question 50

Engineering Mathematics · Numerical Methods: Algebraic equations, integration by trapezoidal and Simpson rules, ODE methods · 2 marks · Numerical answer

The initial value problem $\frac{dy}{dt} + 2y = 0$, $y(0) = 1$ is solved numerically using the forward Euler's method with a constant and positive time step of $\Delta t$.
Let $y_n$ represent the numerical solution obtained after $n$ steps. The condition $|y_{n+1}| \le |y_n|$ is satisfied if and only if $\Delta t$ does not exceed _____________.
(Answer in integer)

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Correct answer: 1

Explanation

The forward Euler step is $y_{n+1} = y_n + \Delta t(-2y_n) = (1 - 2\Delta t)y_n$. The condition $|y_{n+1}| \le |y_n|$ needs $|1 - 2\Delta t| \le 1$, that is $0 \le \Delta t \le 1$. So $\Delta t$ must not exceed 1.