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Unformatted text preview: 8 MA 36600 FINAL REVIEW The general solution is
y(t) = Φ(t) c + Φ(t) t Φ(τ )−1 h(τ ) dτ =⇒ x(t) = T y(t) in terms of the fundamental matrix rt
0 Φ(t) = exp (D t) = .
. ··· ern t . • Consider the nonhomogeneous system
x = P(t) x + g(t).
Denote the matrix
P(τ ) dτ
Q = P(t).
If P and Q commute i.e., if P Q = Q P, then the general solution is
x(t) = Ψ(t) c + Ψ(t)
Ψ(τ )−1 g(τ ) dτ
where Ψ = exp Q(t) is a fundamental matrix.
• In general, say that we can ﬁnd a nonsingular matrix Ψ satisfying
Ψ = P(t) Ψ.
To ﬁnd a particular solution x(p) (t) just with the knowledge of Ψ(t), make a guess that this solution
is in the form x(p) = Ψ(t) u(t) for some function u(t) to be found. Then
u = g(t)
Ψ(τ )−1 g(τ ) dτ .
This is known as the method of Variation of Parameters. ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.
- Spring '09