To solve the differential equation using the Euler$-$Cauchy method, we will follow these steps:

Step $1$: Define the differential equation and the initial condition

The differential equation is:

$d\frac{}{d}t=-t$

with the initial condition:

at $t=0,=2$

Step $2$: Choose the step size $($h$)$ and the range of the solution

The step size is given as $h=0\mathrm{.}1,$ and the range of the solution is from $t=0$ to $t=0\mathrm{.}5.$

Step $3$: Apply the Euler$-$Cauchy method

We will use the Euler$-$Cauchy formula: