final_review (dragged) 7 - 8 MA 36600 FINAL REVIEW The...

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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 e1 0 Φ(t) = exp (D t) = . . . 0 0 er2 t . . . 0 ··· ··· .. . 0 0 . . . ··· ern t . • Consider the nonhomogeneous system d x = P(t) x + g(t). dt Denote the matrix ￿t d Q(t) = P(τ ) dτ =⇒ Q = P(t). dt If P and Q commute i.e., if P Q = Q P, then the general solution is ￿t 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 d Ψ = P(t) Ψ. dt 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 ￿t d Ψ(t) · u = g(t) =⇒ u(t) = Ψ(τ )−1 g(τ ) dτ . dt 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.

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