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11.2 Higher-Order Equations and Systems 477

12. Verify the numerical results given in the text for the system of differential equations (1)
from programs Test RK4 System1 and RK4 System2.
13. (Continuation) Using mathematical software such as Matlab, Maple, or Mathematica
containing symbolic manipulation capabilities to verify the analytic solution for the
system of differential equations (1).
14. (Continuation) Use mathematical software routines such as are found in Matlab, Maple,
or Mathematica to verify the numerical solutions given in the text. Plot the result-
ing solution curve. Compare with the results from programs Test RK4 System1 and
Test RK4 System2.

11.2 Higher-Order Equations and Systems

Consider the initial-value problem for ordinary differential equations of order higher than 1.
A differential equation of order n is normally accompanied by n auxiliary conditions. This
many initial conditions are needed to specify the solution of the differential equation pre-
cisely (assuming certain smoothness conditions are present). Take, for example, a particular
second-order initial-value problem

2

x (t) =−3 cos (t) + 2
(1)
x(0) = 0 x (0) = 0

Without the auxiliary conditions, the general analytic solution is
1 2 3
x(t) = t + cos(2t) + c 1 t + c 2
4 8
where c 1 and c 2 are arbitrary constants. To select one specific solution, c 1 and c 2 must be
3
fixed, and two initial conditions allow this to be done. In fact, x(0) = 0 yields c 2 =− ,
8
and x (0) = 0 forces c 1 = 0.

Higher-Order Differential Equations

In general, higher-order problems can be much more complicated than this simple example
because System (1) has the special property that the function on the right-hand side of the
differential equation does not involve x. The most general form of an ordinary differential
equation with initial conditions that we shall consider is

x (n) = f (t, x, x , x ,..., x (n−1) )

(2)

x(a), x (a), x (a),..., x (n−1) (a) all given

This can be solved numerically by turning it into a system of first-order differential equa-
tions. To do so, we define new variables x 1 , x 2 ,..., x n as follows:
x 1 = x x 2 = x x 3 = x ... x n−1 = x (n−2) x n = x (n−1)