Matrix Exponential: Putzer Formula Variation of Parameters for Systems ...
Matrix Exponential: Putzer Formula Variation of Parameters for Systems Undetermined Coefficients for Systems
? The 2 ? 2 Matrix Exponential eAt ? Putzer Formula for 2 ? 2 Matrices ? How to Remember Putzer's 2 ? 2 Formula
? Variation of Parameters ? Undetermined Coefficients
The 2 ? 2 Matrix Exponential eAt The matrix eAt has columns equal to the solutions of the two problems
u1(t)
=
Au1(t),
u1(0) =
1 0
u2(t)
=
Au2(t),
u2(0) =
0 1
Briefly, the matrix (t) = eAt satisfies the two conditions
1. (t) = A(t),
2. (0) =
10 01 .
Putzer Formula for 2 ? 2 Matrices
eAt
=
e1tI
+
e1t 1
- -
e2t (A
2
-
1I )
eAt = e1tI + te1t(A - 1I)
eAt
=
eat
cos
bt
I
+
eat
sin
bt (A
-
aI )
b
A is 2 ? 2, 1 = 2 real.
A is 2 ? 2, 1 = 2 real.
A is 2?2, 1 = 2 = a+ib, b > 0.
How to Remember Putzer's 2 ? 2 Formula
The expressions (1)
eAt = r1(t)I + r2(t)(A - 1I),
r1(t) = e1t,
e1t - e2t r2(t) = 1 - 2
are enough to generate all three formulas. Fraction r2 is the d/d-Newton quotient for r1. It has limit te1t as 2 1, therefore the formula includes the case 1 = 2 by
limiting. If 1 = 2 = a + ib with b > 0, then the fraction r2 is already real, because it has for z = e1t and w = 1 the form
z - z sin bt
r2(t) = w - w =
. b
Taking real parts of expression (1) gives the complex case formula.
Variation of Parameters
Theorem 1 (Variation of Parameters for Systems)
Let A be a constant n ? n matrix and F(t) a continuous function near t = t0. The unique solution x(t) of the matrix initial value problem
x (t) = Ax(t) + F(t), x(t0) = x0,
is given by the variation of parameters formula
t
(2)
x(t) = eAtx0 + eAt e-rAF(r)dr.
t0
................
................
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