1550078481-Ordinary_Differential_Equations__Roberts_

82 Ordinary Differential Equations

without having to calculate higher order derivatives. As a matter of fact, an
approximation technique is said to be of order m, if the local discretization er-
ror is proportional to the local discretization error of a Taylor series expansion
of order m. Hence, when developing a numerical approximation scheme, the
object is to produce an error which is proportional to hm+l without having
to compute any derivatives of f(x, y).

Euler's Method Again consider the IVP (1) y' = f(x, y); y(xo) = YO·
Substituting the initial condition y(x 0 ) = Yo into the differential equation of
(1), we can calculate y'(xo) = f(xo, y(xo)) = f(xo, Yo), which is the slope of
the tangent line of the exact solution <f;(x) at x = x 0. If in equation

(2) y(x) =Yo+ 1x f(t, y(t)) dt
XO

which we obtained from (1) earlier by integration, we approximate y(t) by
the constant function y 0 and f(t, y(t)) by the constant function f(xo, Yo) on
the interval [x 0 , x 1 ], then we obtain t he following approximation to the exact
solution at x1 :

1

Xl

Y1 =Yo+ f(xo, Yo) dt =Yo+ f(xo, Yo)(x1 - xo).

XO

Knowing y 1 we can compute f(x 1 , y 1 ) from the differential equation in (1).
The value f(x 1 , y 1 ) is an approximation to the slope of the tangent line of the

exact solution <f;(x) at x = x 1. Notice that f(x 1 ,y 1 ) is only an approximation

to <f;'(x 1 ) = f(x 1 , <f;(x 1 )), the slope of the tangent line of the exact solution

<f;(x) at x = x1, since in general Y1 -/= <f;(x 1 ). See Figure 2.12.

Tangent line through (x 0 , YJ

Slope f (x 0 , YJ

Tangent line through (x 1 , $(x 1 ))

Slope f (x 1 , $(x 1 ))

/Exact solution through (x 1 , Y 1 )

/

Figure 2.12 Euler's Method