A better approximation method can be obtained if the integrand in
Eq. (11)
is approximated more accurately. One way to do this is to replace the
integrand by the average of its values at the two endpoints, namely,
. This is equivalent to
approximating the area under the curve between 
and 
by the area of the shaded trapezoid. Further, we replace 
and
by their respective approximate values 
and 
in this way we obtain from Eq. (11)
Since the unknown 
appears as one of the arguments of f on the
right side of Eq. (12), this equation defines 
implicitly
rather than explicitly. Depending on the nature of the function f,
it may be fairly difficult to solve Eq. (12) for 
.
The difficulty can be overcome by replacing 
on the right side of
Eq. (12) by the value obtained using the Euler formula. Thus
where 
has been replaced by 
.
The above equation gives an explicit formula for computing 
,
the approximate value of 
, in terms of the data at 
.
This formula is known as the improved Euler formula or the
Heun formula. The improved Euler formula is an example of a
two-stage method; that is, we first calculate 
from the
Euler formula and then use this result to calculate 
. The
local truncation error for the improved formula is 
as opposed to
for the Euler's method. It can also be shown that for a finite
interval, the global truncation error for the improved Euler formula
is bounded by 
, so this method is a second order method. However, this
accuracy is achieved at the expense of more computational work.
What happens when 
depends only on t andn not on y?
The differential equation 
reduces to integrating 
.
In this case, the improved Euler formula becomes
which is the trapezoid rule for numerical integration.
