74 Ordinary Differential Equations
In the late nineteenth century and early twentieth century, several com-
mercially viable mechanical calculators capable of adding, subtracting, multi-
plying, and dividing were invented and manufactured. Electric motor driven
calculators b egan to appear as early as 1900. These mechanical and electrical
computing devices improved the speed and accuracy of generating numerical
solutions of simple differential equations. In 1936, the German civil engineer
Konrad Zuse (1910-1995) built the first mechanical binary computer, the Zl,
in the living room of his parents' home. From 1942 to 1946 the first large
scale, general purpose electronic computer was designed and built by John
W. Mauchly (1907-1980) and J. Presper Eckert (1919-1995) at the University
of Pennsylvania. The computer was named ENIAC, which is an acronym
for "Electronic Numerical Integrator and Computer." ENIAC, which used
vacuum tube technology, was operated from 1946 to 1955. After many tech-
nological inventions such as the transistor and integrated circuitry, the first
hand-held, battery-powered, pocket calculator capable of performing addition,
subtraction, multiplication, and division was introduced by Texas Instruments
in 1967. The first scientific pocket calculator, the HP-35, was produced in 1972
by Hewlett Packard.
In the 1960s and 1970s several sophisticated computer programs were de-
veloped to numerically solve differential equations. Since then significant ad-
vances in graphical display capabilities have occurred also. Consequently, at
the present time there are many computer software packages available to gen-
erate numerical solutions of differential equations and to graphically display
the results.
In this section we present some of the simpler single-step, multistep, and
predictor-corrector methods for computing numerical approximations to the
solution of the initial value problem
(1) y' = f(x, y); y(xo) =Yoand for estimating the error of the computed approximations. We discuss the
advantages and disadvantages of each type of method. Then we present and
discuss desirable features for computer software to solve the IVP (1). Next,
we explain how to use computer software to generate a numerical approxi-
mation to the solution of the first-order IVP (1). Finally, we illustrate and
interpret the various kinds of results which computer software may produce.
Furthermore, we reiterate the importance of performing a thorough mathe-
matical analysis, which includes applying the fundamental theorems to the
problem, prior to generating a numerical approximation.
Before we start generating numerical approximations to the solution of the
IVP (1), we need to have a basic understanding of how a digital computer
represents and processes numbers. First of all, it is impossible to represent all
real numbers exactly in a digital computer. In most cases, a digital computer
represents and stores real numbers as floating-point quantities using a scheme
similar to scientific notation. Since a digital computer is a finite device, only