84 Ordinary Differential Equationsequation (12) is an upper bound and not a realistic error estimate. However,
equation (12) does show that as h ---> 0, the global discretization error goes
to zero. Thus, neglecting round-off error, as h---> 0, the Euler approximation
converges to the exact solution.
The Swiss mathematician Leonhard Euler (1707-1783) was one of the most
prolific authors in the field of mathematics. He introduced the now common
notations of e for the base of natural logarithms, n for the ratio of the cir-
cumference of a circle to its diameter, :E for the sign for summation, i for the
imaginary unit, and sin x and cos x for the trigonometric functions. Euler'sfamous formula ei!J = cos e + i sin e expresses a relation between the exponen-
tial function and trigonometric functions. The special case e = n provides the
following relation between the numbers e, 7r, and i: ei7r = -1. Euler's contri-
bution to differential equations includes the concept of an integrating factor,
the method for solving linear differential equations with constant coefficients,
the method of reduction of order, and power series solutions, to name a few.EXAMPLE 3 Euler's Approximation to the Solution of the
IVP: y' = y + x; y(O) = 1
a. Find an approximate solution to the initial value problem(7) y' = y + x = J(x, y); y(O) = 1
on the interval [O, 1] using Euler's method and constant stepsizes h = .2 and
h = .1.
b. Use equation (12) to estimate the maximum global discretization error at
x = 1 for constant stepsizes h = .2 and h = .1.SOLUTION
a. Table 2.1 contains Euler's approximation to the IVP (7) on the interval [O, 1]obtained using a constant stepsize of h = .2. And Table 2.2 contains Euler's
approximation to the IVP (7) on the interval [O, 1] obtained using a constantstepsize of h = .1. All calculations were performed using six significant digits.
b. Taking the partial derivative of f(x, y) = y + x with respect to y, we find
fy = 1, so lfyl :S 1 = L on [O, l]. Differentiating the differential equation
a ppearing in (7), y' = y + x, we get y(^2 ) = y' + 1 = y + x + l. Assuming, as we
did in the Taylor series expansion example, that jy(x)I < 7 for x E [O, 1], we
have jy(^2 )1 < IYI + lxl + 1 < 9 = Y for x E [O, l]. Therefore, by equation (12)
with h = .2 the maximum global discretization error satisfies
IYs - </>(l)j :S ~~ (e(xn-xa)L - 1) = (·
2
;(
9
) (e - 1) ~ 1.54645.