2130_Chapter7_Notes

Bernard fulgham university of virginia july 23 2013

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Unformatted text preview: 1 (t0 ) Ψ(t0 ) c = Ψ−1 (t0 ) x0 =⇒ I c = Ψ−1 (t0 ) x0 =⇒ c = Ψ−1 (t0 ) x0 . 3 / 12 2 Special Fundamental Matrix Define Φ(t) = Ψ(t) Ψ−1 (t0 ), then the solution of the IVP x′ = P(t) x, x(t0 ) = x0 is: x = Ψ(t) Ψ−1 (t0 ) x0 = Φ(t) x0 . Φ(t) Note: Φ(t) is the unique fundamental matrix with Φ(t0 ) = I. Theorem If Ψ(t) is any fundamental matrix for x′ = P(t) x, then the solution with initial condition x(t0 ) = x0 is: x = Φ(t) x0 = Ψ(t) Ψ−1 (t0 ) x0 . 4 / 12 Example 1 x′ = 11 x 41 Fundamental set of solutions: x(1) = 1 3t e, 2 x(2) = Fundamental matrix: Ψ(t) = x(1) x(2) = 1 −t e. −2 e3t e−t 3t −2e−t . 2e Next, compute the special fundamental matrix Φ(t) at t0 = 0. 5 / 12 3 Exa 1 (slide 2) Fundamental matrix: Ψ(t) = x(1) x(2) = e3t e−t 3t −2e−t . 2e Special fundamental matrix: Φ(t) = Ψ(t)Ψ−1 (0). 1 1 2 −2 Ψ−1 (0) = =⇒ Φ(t) = −1 =− 12 1 −2 −1 1 = 1 4 −2 4 2 −1 1 e3t e−t 3t −2e−t 4 2e 2 1 2 −1 = 1 2(e3t...
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This note was uploaded on 03/19/2014 for the course APMA 2130 taught by Professor Bernardfulgham during the Fall '09 term at UVA.

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