An explicit forward Euler method is used to numerically integrate the differential equation 

using a time step of 0.1. With the initial condition y(0) = 1 , the value of y(1) computed by this method is ___________ (correct to two decimal places).

The correct answer is 2.59.

Step-by-Step Explanation:

We are given the ordinary differential equation:
$\frac{dy}{dt}=y$
with the initial condition $y\left(0\right)=1$ and a time step of $h=0.1$.

The explicit forward Euler method is defined by the iteration formula:
$y_{n+1}=y_n+hf(t_n,y_n)$
where $f(t,y)=y$ in this case.

Substituting the given function and step size into the formula, we obtain:
$y_{n+1}=y_n+0.1y_n=1.1y_n$

We want to find the value of $y\left(1\right)$. The number of steps $N$ required to go from $t=0$ to $t=1$ with $h=0.1$ is:
$N=\frac{1-0}{0.1}=10$

Therefore, after 10 iterations, the numerical solution is:
$y_{10}={\left(1.1\right)}^{10}{y}_0$

Given the initial condition $y_0=y\left(0\right)=1$, we calculate:
$y_{10}={\left(1.1\right)}^{10}\times 1\approx 2.5937$

Rounding to two decimal places, we get:
$y\left(1\right)\approx 2.59$