
EXERCISES
3.8 Referring to Figure 3.7,
(a) Find the equation of the linear approximation u
1
(t) through the end points [nT, u(n)] and
[(n þ1)T, u(n þ1)].
(b) Verify Equation 3.39 by integrating u
1
(t) from nT to (n þ1)T.
3.9 The first-order system dx=dt ¼lx with initial condition x(0) ¼x
0
is to be simulated using
trapezoidal integration with step size T. The truncation error after n steps is e
n
¼x
A
(n)
x(nT), where x(t), t 0 is the exact solution and x
A
(n), n ¼0, 1, 2, . . . is the approximate
(simulated) solution, that is, x
A
(n) x(nT), n ¼0, 1, 2, 3, . . . . Suppose the truncation error after
the first step is equal to a fraction of the initial condition, ...