1550078481-Ordinary_Differential_Equations__Roberts_

The Initial Value Problem y' = f(x, y); y(c) = d 77

Observe that the absolute error at Xi is the vertical distance between the

points (xi, ¢(xi)) and (xi, Yi)· See Figure 2.11.

Most differential equations and initial value problems cannot be

solved explicitly or implicitly; and, therefore, we must be satisfied

with obtaining a numerical approximation to the solution. Thus,

we need to know how to generate a numerical approximation to the solution

of the IVP (1) y' = f(x, y); y(xo) = y 0. In the differential equation in (1)

replace y' by dy/ dx, multiply by dx, and integrate both sides of the resulting

equation from xo to x to obtain

r dy = r f(t, y(t)) dt or y(x) - y(xo) = r f(t, y(t)) dt.

lxo lxo lxo

Adding y(xo) = Yo to the last equation , we find the symboli c solution to t h e

IVP (1) on [xo, x] to be

(2) y(x) =Yo+ r f(t, y(t)) dt.

lxa

When f(x, y) in (1) is a function of the independent variable alone-that
is, when the initial value problem is y' = f(x); y(xo) =yo- we can approx-
imate the function f(x) on the interval [xo, x 1 ], where x 1 is a specific point,
by step functions or some polynomial in x, say p 1 ( x), and then using this
approximation integrate (2) over [xo, x 1 ] to obtain an approximation y 1 to
the solution ¢(x 1 ). Next, we approximate f(x) on [x 1 , x2] by some function

P2(x) and integrate over [x1, x2] to obtain Y2 = Y1 + fxx 12 p2(t) dt which is an

approximation to the solution ¢(x 2 ), and so on.

When f(x, y) in (1) is a function of the dependent variable y, the value of
the approximate solution y 1 at x 1 depends on the unknown solution ¢(x) on
the interval [xo, x 1 ] and the function f(x, y) on the rectangle

R1={(x,y) l xo:::;x:::;x1, yE{¢(x)lxo:::;x:::;x1}}.

Thus, we must approximate ¢(x) on [xo, x1] and f(x, y) on R1 in order to
be able to integrate (2) over the interval [x 0 , x 1 ] and obtain a n approximate

solution y 1. In this case, additional approximate values Y2, ... , Yn are obtained

in a like manner.

Now suppose we have generated the numerical approximations (xo, Yo),
(x 1 , y1), ... , (xn, Yn)· If the algorithm for generating the numerical approx-
imation at Xn+i depends only on (xn, Yn) and our approximation of f (x, y)
at (xn, Yn), then the algorithm is call ed a single-step, one-step, stepwise,

or starting method. On the other hand, if the algorithm for generating

the numerical approximation at Xn+l depends on (xn, Yn), (Xn-1, Yn-1), ... ,
(Xn-m, Yn-m) where m 2::: 1, then the procedure for generating the numerical

approximation is known as a multistep or continuing method.