Question

The ordinary differential equation $\frac{dx}{dt}=-3x+2$,with $x\left(0\right)=1$ is to be solved using the forward Euler method. The largest time step that can be used to solve the equation without making the numerical solution unstable is

• A. 0.66

Answer 1

The correct option is A 0.66
$\frac{dx}{dt}=-3x+2=f(t,x)$ & $x\left(0\right)=1$
Iterative equation is:
$x_{n+1}=x_n+hf(t_n,x_n)$
$=x_n+h[-3x_n+2]$
$x_{n+1}=(1-3h)x_n+2h$
So method will be statbe only when $|1-3h|\le 1$
$-1\le 1-3h\le 1$
$-2\le -3h\le 0$
$0\le +3h\le 2$
$0\le h\le \frac{2}{3}$
$\therefore$ Maximum value of $h=\frac{2}{3}=0.66$