Most ordinary differential equations arising in
real-world applications

cannot be solved exactly.
These ode can be analyized qualitatively.

However,
qualitative analysis may not be able to give accurate
answers.

A *numerical method* can be used to get
an accurate approximate solution

to a differential
equation. There are many programs and packages
for

solving differential equations. With today's
computer, an accurate

solution can be obtained
rapidly. In this section we focus on Euler's

method,
a basic numerical method for solving initial
value problems.

Consider the differential equation:

The first step is to convert the above second-order
ode into two

first-order ode. This is a
standard operation. Let v(t)=y'(t).

Then v'(t)=y''(t).
We then get two differential equations. The

first
is easy

The second is obtained by rewriting the original
ode. Using the fact

that y''=v' and y'=v,

The initial conditions are y(0)=1 and y'(0)=v(0)=2.

We are now ready to approximate the two first-order
ode by Euler's

method. A derivation
of Euler's method
is given the numerical methods

section for first-order
ode. We first discretize the time interval.

Let
0=t_0. Let t_1<t_2<t_3< ... be other points.
We approximate

the solution at these gridpoints.
For simplicity we will assume that

the points are
equispaced. In general they need not be.
Let us define

z_k to be the approximation to y(t) at t_k
and let w_k be the

approximation to v(t) at t_k. For
our example we get the following

recursion formulas for
z_k and w_k:

The initial conditions are z_0=y(0)=1 and w_0=v(0)=2. Here Dt is the

spacing
between gridpoints.

Suppose our goal is to compute the solution to the model
differential

equation at time t=1. The exact solution,
obtained using an advanced

algorithm, is 4.1278. The
following table summarizes the results obtained

using
Euler's method. Note that the error decreases as
the number

of gridpoints N increases. Here
the spacing between points is 1/N.

**Summary**

Consider the second-order ode:

Suppose the goal is solve the problem on the interval
[t_0,T]. The

following list summarizes the steps in
solving the problem by Euler'

method.

- Convert the second-order ode into two first-order
ode. Let

v=y'. Then the two odes are - Discretize the interval [t_0,T]. Pick a bunch of points

t_0<t_1<t_2<...<t_N=T. The points need not be equispaced. - Let z_k denote the approximation to y(t_k) and let
w_k denote the

approximation to v(t_k). - Use the formulas
to compute the approximate solution for k=0,1,2,...

- Netlib: This is a repository
for all sorts of mathematical software.

Most of the programs are in C or Fortran. Look under ode or odepack. - Maple, Mathematica,
Matlab: These are packages for doing
numerical

and symbolic computations. They have routines for solving ode

numerically.

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**Copyright **© **1996
Department
of Mathematics, Oregon State
University**

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