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466 Chapter 11 Systems of Ordinary Differential Equations

Taking an example of more modest scope, we consider two equations with two auxiliary
conditions. Let x and y be two functions of t subject to the system

2
x (t) = x(t) − y(t) + 2t − t − t 3

(1)
2

y (t) = x(t) + y(t) − 4t + t 3
with initial conditions

x(0) = 1
y(0) = 0
This is an example of an initial-value problem that involves a system of two first-order
differential equations. Note that in the example given, it is not possible to solve either of
the two differential equations by itself because the first equation governing x involves the

unknown function y, and the second equation governing y involves the unknown function x.

In this situation, we say that the two differential equations are coupled.
The reader is invited to verify that the analytic solution is
t 2 2
x(t) = e cos(t) + t = cos(t)[cosh(t) + sinh(t)] + t
3
t
y(t) = e sin(t) − t = sin(t)[cosh(t) + sinh(t)] − t 3
Let us look at another example that is superficially similar to the first but is actually
simpler:

2
x (t) = x(t) + 2t − t − t 3

(2)
2
y (t) = y(t) − 4t + t 3

with initial conditions

x(0) = 1
y(0) = 0
These two equations are not coupled and can be solved separately as two unrelated initial-
value problems (using, for instance, the methods of Chapter 10). Naturally, our concern
here is with systems that are coupled, although methods that solve coupled systems also
solve those that are not. The procedures discussed in Chapter 10 extend to systems whether
coupled or uncoupled.
Taylor Series Method

We illustrate the Taylor series method for System (1) and begin by differentiating the
equations constituting it:
2 3
x = x − y + 2t − t − t

2
y = x + y − 4t + t 3

2
x = x − y + 2 − 2t − 3t

y = x + y − 8t + 3t 2

x = x − y − 2 − 6t

y = x + y − 8 + 6t
(4)
x = x − y − 6

y (4) = x + y + 6

etc.