To derive the error, again, we simply use Taylor series, but in an appropriate form.
Instead of approximating for y(*t* + *h*) given y(*t*), we approximate for y(*t*) given
y(*t* + *h*):

Note that the second term is negative because *t* = (*t* + *h*) − *h*.

From the differential equation, we may substitute y^{(1)}(*t*) = f(*t*, y(*t*)):

If we substitute y(*t*) = *y*_{0}, y(*t* + *y*) = *y*_{1},
and *t* + *h* = *t*_{1} into this equation, we get:

If we remove the *h*^{2} term, we get the equation which we
are solving for the backward-Euler's method. With further manipulation, it
can be argued that this implies that the error also drops with respect to
*h*^{2}.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.