2130_Chapter7_Notes

# Bernard fulgham university of virginia july 23 2013

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 03/19/2014 for the course APMA 2130 taught by Professor Bernardfulgham during the Fall '09 term at UVA.

Ask a homework question - tutors are online