Lecture Notes in Differential Equations - Bruce E. Shapiro

304 LESSON 31. LINEAR SYSTEMS
a fundamental set of solutions if every solution to (31.4) can be written
in the form
y = C 1 y 1 (t) + · · · + C n y n (t) (31.5)
for some set of constants C 1 , . . . , C n . We define the fundamental matrix
as
W = ( )
y 1 · · · y n (31.6)
and the Wronskian as its determinant,
W (t) = det W (31.7)
The columns of the fundamental matrix contain the vector-valued solutions,
not a solution and its derivatives, as they did for the scalar equation.
However, this should not be surprising, because if we convert an nth order
equation into a system by making the change of variables u 1 = y, u 2 = y ′ ,
u 3 = y ′′ , ..., u n = y (n−1) , the two representations will be identical.
Homogeneous Systems
Theorem 31.1. The fundamental matrix of the system y ′ = Ay satisfies
the differential equation.
Proof. Let y 1 , ..., y n be a fundamental set of solutions. Then each solution
satisfies y i ′ = Ay i. But by definition of the fundamental matrix,
W ′ = ( )
y 1 ′ · · · y n
′ (31.8)
= ( )
Ay 1 · · · A]y n (31.9)
= A ( )
y 1 · · · y n (31.10)
Thus W satisfies the same differential equation.
= AW (31.11)
For n = 2, we can write equation the homogeneous system as
}
u ′ = au + bv
v ′ = cu + dv
where u = y 1 , v = y 2 , and a, b, c, and d are all real constants.
the characteristic equation of the matrix
( ) a b
A =
c d
(31.12)
(31.13)