2. Linear Systems of Ordinary Differential Equations

Suppose that an n×n matrix function A(t) is continuous for 0 ≤ t ≤ b, b > 0. for any vector c, consider the initial value problem

dx/dt = A(t) x

(1)

x(0) = c

(2)

There exists, for a given c, a value b₀ such that (1) and (2) have a unique solution for 0 ≤ t ≤ b₀. The proof of this theorem also shows that if A(t) can be extended to -b ≤ t ≤ b, we have a solution for -b₁ ≤ t ≤ 0; that is, we can obtain a unique solution to (1) and (2) for -b₀ ≤ t ≤ b₀, where 0≤ b₀ ≤ b.

Lemma 1. If x(t) satisfies (1) and (2) on -b₀ ≤ t ≤ b₀, then for any a ∈ R, y(t) = a x(t) satisfies the initial value problem

dy/dt = A(t) y

y(0) = a c

Lemma 2. For i = 1, 2, if x⁽ⁱ⁾(t) satisfies

dx⁽ⁱ⁾/dt = A(t) x⁽ⁱ⁾(t) , x⁽ⁱ⁾(0) = c⁽ⁱ⁾

on -b_i ≤ t ≤ b_i, then y(t) = x¹(t) + x⁽²⁾(t) satisfies

dy/dt = A(t) t(t) , y(0) = c⁽¹⁾ + c⁽²⁾

on -b₀ ≤ t ≤ b₀, where b₀ = min (b₁, b₂).

[The proofs of these lemmas are elementary.]

Since every c can be written as a linear combination c = a₁ c⁽¹⁾ + ... + a_n c⁽ⁿ⁾ where {c⁽¹⁾, ... , c⁽ⁿ⁾} is a basis of Rⁿ, we see that there exists a neighbourhood of the origin in which the solutions of (1) form a vector space of dimension n.

Proposition. If A(t) is continuous on [0,b] the set of solutions of (1) on [0,b] form a vector space of dimension n.

Proof. By Lemmas 1 and 2 and the remarks following equation (2), we see that for any s ∈ [0,b], there exists a neighbourhood N_s of s on which solutions of (1) exist and form an n-dimensional vector space. Given two points s₁, s₂ ∈ [0,b] such that N_s1 and N_s2 intersect, the solutions can be patched up (see Appendix) to give a solution on N_s1 ∪ N_s2. By compactness of [0,b], there exists a set of points s_i ∈ [0,b], i = 1, ... , M such that N_s1 ∪ ... ∪ N_sM = [0,b]. Putting together the solutions on each N_si, we obtain a vector space of solutions on the whole interval.

Linear systems with constant coefficients. Consider the system

dx/dt = A x

(3)

where A is a constant matrix. The infinite series

exp(tA) = ∑_k=1,...,∞t^kA^k/k!

converges for all values of t and A. Furthermore, it is easy to see, by differentiation of power series, that

(d/dt)[exp(tA)] = A exp(tA)

(*)

If t = 0, exp(tA) = I, the identity matrix. Multiplying (*) by any vector c, we then see that

x(t) = exp(tA) c

(4)

satisfies the initial value problem

dx/dt = A x , x(0) = c

(5)

Higher order linear differential equations. The linear differential equation

a₀(t) x + a₁(t) dx/dt + a₂(t) d²x/dt² + ... + a_n(t) dⁿx/dtⁿ = 0

(6)

can be written as

x₁ = x , dx₁/dt = x₂ , dx₂/dt = x₃ , ... , dx_n-1/dt = x_n,

b₁(t) dx₁/dt + ... + b_n(t) dx_n/dt = x₁

where

b_i(t) = - a_i(t)/a₀(t) , i = 1, ... , n

(7)

or

B(t) dx/dt = x

that is

dx/dt = B^-1(t) x

(8)

where x is the vector (x, dx/dt, ... , d^n-1x/dt^n-1) and

(9)

A solution of (8) exists in any interval where a₀(t) ≠ 0. Thus the higher order differential equation (6) has been reduced to a first order system of linear equations (8).

References. This page only develops the most elementary part of the theory of linear equations, which will be needed in the sequel. Further information may be found in the following books:

1. Theory of ordinary differential equations -- Earl A. Coddington and Norman Levinson
2. Nonlinear Differential Equations -- Raimond A. Struble

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