MATH 164, Spring 2001 Due Date: Name(s):
Extra Project 10.2b: Euler's Method
Objective
To investigate Euler's method.
Narrative
If you have not already done so, read Section 10.2 of the text. The key ideas you should take away from
this section are:
1. Not all differential equations can be solved in closed form. There are techniques, however, such as
Euler's Method which allow you to obtain the approximate value of a solution at a given point.
2. For small values of h we can approximate
f (x) = lim
h0
f(x + h) - f(x)
h

f(x + h) - f(x)
h
. (1)
3. Given an interval [a, b] and an integer N, if xn = a + nh, where n = 0, . . . , N and h = (b - a)/N, then
equation (1) implies that
f (xn) 
f(xn+1) - f(xn)
h
or f(xn+1)  f(xn) + hf (xn),
or, letting yn = f(xn),
yn+1 = yn + hf (xn). (2)
Task
a) Type the command lines below into Maple in the order in which they are listed. These commands produce
the direction field associated to the differential equation y = x + y.
> # Project 10.2b: Euler's Method
> restart: with(DEtools):
> y0 := y(x); y1 := diff(y(x),x);
> dfieldplot(y1 = x+y0,y0,x=-4..4,y=-4..4);
b) Continue by typing the following command lines into Maple in the order in which they are listed. These
commands solve y = x + y with the initial condition y(0) = 1, analytically, and evaluate this solution when
x = 1.
> dsolve({y1 = x+y0,y(0)=1},y0);
> evalf(subs(x=1,%));
c) Continue by typing the following command line into Maple. This command produces the direction field
associated to the differential equation y = x + y at a finer scale.
> dfieldplot(y1 = x+y0,y0,x=0..1,y=1..4);
d) Continue by typing the following command lines into Maple in the order in which they are listed. These
commands apply Euler's Method (equation (2)) to y = x + y assuming that a = 0, b = 1, and N = 10.
(That is, they use equation (2) to approximate y(1) if y = x + y and y(0) = 1 using N = 10 steps.)
> a := 0.0; b := 1.0; N := 10;
> x.0 := a; y.0 := 1.0; h := (b-a)/N;
> for n from 0 to N-1 do
y.(n+1) := y.n + h * (x.n + y.n ): x.(n+1) := x.n + h:
print(n, x.(n+1), y.(n+1)); od;
Note the discrepancy between the analytically computed value of y(1) and the numerical value computed
with Euler's Method.
e) Repeat part (d) using N = 100 subintervals. (The easiest way to do this is to copy and paste the lines
you typed in part (d), to change N from 10 to 100, and then to "re-Enter" the commands.) Note how
much closer the numerical value computed with Euler's Method is to the exact value (which we computed
analytically).
At this point, make a hard-copy of your typed input and Maple's responses (both text and graphics).
Then, ...
f) On the direction field you produced in part (c), use the data you generated in part (d) to draw the sequence
of line segments from (xn, yn) to (xn+1, yn+1), n = 0, . . . , 9, which graphically embodies Euler's Method.
Comments
1. In this project, we solved a differential equation analytically, graphically, and numerically, and were able
to compare our solutions. In many cases, it is difficult -- or even impossible -- to solve a differential
equation analytically, and in this case the only alternatives are graphical and numerical solutions.
2. While Euler's Method is good for illustrating the general approach to numerically solving differential
equations, it does not generally yield extremely accurate results. Other numerical techniques exist, and
these are studied in courses on Numerical Analysis.