Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 31
Linear Systems
The general linear system of order n can be written as
y ′ 1 = a 11 y 1 + a 12 y 2 + · · · + a 1n + f 1 (t)
⎫
⎪⎬
.
⎪⎭
y n ′ = a n1 y 1 + a n2 y 2 + · · · + a nn + f 2 (t)
(31.1)
where the a ij are either constants (for systems with constant coefficients) or
depend only on t and the functions f i (t) are either all zero (for homogeneous
systems) or depend at most on t.
We typically write this a matrix equation
⎛ ⎞ ⎛
⎞ ⎛ ⎞ ⎛ ⎞
y 1
′ a 11 a 12 · · · a 1n y 1 f 1 (t)
⎜ ⎟ ⎜
⎝ . ⎠ = ⎝
.
. ..
⎟ ⎜ ⎟ ⎜ ⎟
. ⎠ ⎝ . ⎠ + ⎝ . ⎠ (31.2)
a n1 · · · a nn y n f n (t)
y ′ n
We will write this as the matrix system
y ′ = Ay + f (31.3)
where it is convenient to think of y, f(t) : R → R n as vector-valued functions.
In analogy to the scalar case, we call a set of solutions to the vector equation
{y 1 , ..., y n } to the homogeneous equation
y ′ = Ay (31.4)
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