Suppose that we use the Improved Euler's method to approximate the solution to the differential equation $$\frac{dy}{dx}= x - 0.5 y;\qquad y( 0.1) = 6.$$ Let $f(x,y) =x-0.5 y.$
We let $x_0 = 0.1$ and $y_0 = 6$ and pick a step size $h=0.25.$ The improved Euler method is the the following algorithm. From $(x_n, y_n ),$ our approximation to the solution of the differential equation at the $n$-th stage, we find the next stage by computing the $x$-step $x_{n+1}=x_n+h,$ and then $k_1,$ the slope at $(x_n,y_n).$ The predicted new value of the solution is $z_{n+1} = y_n + h\cdot k_1.$ Then we find the slope at the predicted new point $k_2 = f(x_{n+1}, z_{n+1})$ and get the corrected point by averaging slopes $$y_{n+1} = y_n + \frac h2 (k_1 + k_2).$$ Complete the following table:

$n$	$x_n$	$y_n$	$k_1$	$z_{n+1}$	$k_2$
$0$	$0.1$	$6$	$-2.9$	$5.275$	$-2.2875$
$1$
$2$
$3$
$4$

The exact solution can also be found for the linear equation. Write the answer as a function of $x$.
$y(x)=$

Thus the actual value of the function at the point $x=1.1$ is
$y(1.1) =$ .