Lecture Notes in Differential Equations - Bruce E. Shapiro

308 LESSON 31. LINEAR SYSTEMS
We can find v by substitution int (31.36):
u = 29 5 e2t + 1 5 e−3t (31.43)
v = u ′ − u (31.44)
= 58 5 e2t − 3 5 e−3t − 29
5 e2t − 1 5 e−3t (31.45)
= 29 5 e2t − 4 5 e−3t (31.46)
The Matrix Exponential
Definition 31.3. Let M be any square matrix. Then we define the exponential
of the matrix as
exp(M) = e M = I + M + 1 2 M2 + 1 3! M3 + 1 4! M4 + · · · (31.47)
assuming that the series converges.
Theorem 31.4. The solution of y ′ = Ay with initial conditions y(t 0 ) = y 0
is y = e A(t−t0) y 0 .
Proof. Use Picard iteration:
Then
and in general
Φ 0 (t) = y 0 (31.48)
Φ k (t) = y 0 +
∫ t
t 0
AΦ k−1 (s)ds (31.49)
∫ t
Φ 1 = y 0 + AΦ 0 (s)ds
t 0
(31.50)
⎡
Φ k = ⎣I +
∫ t
= y 0 + Ay 0 ds (31.51)
t 0
= y 0 + Ay 0 (t − t 0 ) (31.52)
= [I + A(t − t 0 )]y 0 (31.53)
k∑
j=1
1
k! Ak (t − t 0 ) k ⎤
⎦ y 0 (31.54)