Answer :

Solution :

Given scheme is

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

$\frac{dy}{dt}=-\pi y$

From the scheme,

⇒ y(t_n+1) – y(t_n) = - πh y(t_n)

⇒ y(t_n+1) = y(t_n) (1 - πh)

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

The scheme will be stable if $\left|\frac{y_{n+1}}{y_n}\right|<1$ , as the series will converge

⇒ |1 – πh| < 1

⇒ -1 < 1 – πh < 1

⇒ - 2 < - πh < 0

⇒ 0 < πh < 2;

⇒ 0 < h < 2/π;