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GATE Mechanical 2021 Set 1 | Question: 4

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The ordinary differential equation $\dfrac{dy}{dt}=-\pi y$ subject to an initial condition $y\left ( 0 \right )=1$ is solved numerically using the following scheme:

$$\frac{y\left ( t_{n+1} \right )-y\left ( t_{n} \right )}{h}=-\pi y\left ( t_{n} \right )$$

where $\text{h}$ is the time step, $t_{n}=nh$, and $n=0,1,2,\dots$. This numerical scheme is stable for all values of $h$ in the interval ___________________.

  1. $0< h< \frac{2}{\pi }$
  2. $0< h< 1$
  3. $0< h< \frac{\pi }{2}$
  4. for all $h> 0$
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