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GATE 2021 ME (ME1) – Question 14

Engineering Mathematics · Numerical Methods: Algebraic equations, integration by trapezoidal and Simpson rules, ODE methods · 1 mark · Multiple choice

The ordinary differential equation $\frac{dy}{dt}=-\pi y$ subject to an initial condition $y(0)=1$ is solved numerically using the following scheme: $\frac{y(t_{n+1})-y(t_n)}h=-\pi y(t_n)$, where $h$ is the time step, $t_n=nh$, and $n=0,1,2,\ldots$. This numerical scheme is stable for all values of $h$ in the interval

  1. $0<h<2/\pi$
  2. $0<h<1$
  3. $0<h<\pi/2$
  4. All h>0

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Correct answer: (A) $0<h<2/\pi$

Explanation

The recurrence is $y_{n+1}=(1-\pi h)y_n$. Perturbations decay only when the amplification factor has magnitude below 1.

Solve $-1<1-\pi h<1$ to get **0<h<2/π (A)**. At the upper endpoint perturbations oscillate without decaying.