2. Linear Ordinary Differential Equations
In this chapter we shall present a systematic study of the solutions of linear
ordinary differential equations. Such equations continue to play an important
role in many branches of the natural sciences and other fields, e. g., economies,
so that they may be treated here in their own right. On the other hand, we should
not forget that our main objective is to study nonlinear equations, and in the construction of their solutions the solutions of linear equations come in at several
instances.
This chapter is organized as follows. Seetion 2.1 will be devoted to detailed
discussions of the solutions of different kinds of homogeneous differential equations. These equations will be distinguished by the time dependence of their coefficients, i. e., constant, periodic, quasiperiodic, or more arbitrary. In Sect. 2.2 we
shall point out how to apply the concept of invariance against group operations
to the first two types of equations. Section 2.3 deals with inhomogeneous differential equations. Some general theorems from algebra and from the theory of
linear ordinary differential equations (coupled systems) will be presented in Sect.
2.4. In Sect. 2.5 dual solution spaces are introduced. The general form of solution for the cases of constant and periodic coefficient matrices will be treated in
Sects. 2.6 and 2.7, 8, respectively. Seetion 2.8 and the beginning of Sect. 2.7 will
deal with aspects of group theory, Sect. 2.8 also including representation theory.
In Sect. 2.9, aperturbation theory will be developed to get explicit solutions for
the case of periodic coefficient matrices.
2.1 Examples of Linear Differential Equations: The Case of a
Single Variable
We consider a variable q which depends on a variable t, i.e., q(t), and assume,
that q is continuously differentiable. We shall interpret t as time, though in
certain applications other interpretations are also possible, for instance as aspace
coordinate. If not otherwise stated it is taken that - 00 < t< + 00. We consider
various typical cases of homogeneous first-order differential equations.
-2.1.1 Linear Differential Equation witb Constant Coefficient
We consider
q=aq,
H. Haken, Advanced Synergetics
© Springer-Verlag Berlin Heidelberg 1983
(2.1.1)
62
2. Linear Ordinary Differential Equations
where a is a constant. The solution of this differential equation reads
q(t)
= Ce u
(2.1.2)
,
where Ä = a as can be immediately checked by inserting (2.1.2) into (2.1.1). The
constant C can be fixed by means of an initial condition, e. g., by the requirement
that for t = 0
q(O)
= qo,
(2.1.3)
where qo is prescribed. Thus
(2.1.4)
C= qo=q(O).
Then (2.1.2) can be written as
q(t)
= q(O) eU ,
(Ä
= a).
(2.1.5)
As shown in Sects. 1.14, 16, linear differential equations are important to determine the stability of solutions of nonlinear equations~ Therefore, we shall discuss
here and in the following the time dependence of the solutions (2.1.5) for large
times t. Obviously for t > 0 the asymptotic behavior of (2.1.5) is governed by the
sign ofRe{Ä}. For Re{Ä} > 0, Iqlgrows exponentially, for Re{Ä} = 0, q(t) is a constant, and for Re{Ä} < 0, Iq I is exponentially damped. Here Ä itself is called a
characteristic exponent.
2.1.2 Linear Differential Equation with Periodic Coefficient
As a further example we consider
q = a(t)q,
(2.1.6)
where a(t) is assumed continuous. The solution of (2.1.6) reads
q
= q(O) exp
[ta(T)dT].
(2.1.7)
We now have to distinguish between different kinds of time dependence of a(t).
If a(t) is periodic and, for instance, continuously differentiable, we may expand
it into a Fourier series of the form
a(t)
<XI
= Co + L
n= -00
n*O
•
cn e Inwt
(2.1.8)
To study the asymptotic behavior of (2.1.7), we insert (2.1.8) into the integral
occurring in (2.1.7) and obtain