2130_Chapter7_Notes

# x pn1 tx1 pn2 tx2 pnn txn gn t n if

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Unformatted text preview: · + p2n (t)xn + g2 (t), 2 . . . x′ = pn1 (t)x1 + pn2 (t)x2 + · · · + pnn (t)xn + gn (t). n If each gk (t) = 0, then the system is homogeneous; otherwise, it is nonhomogeneous. Every nth order linear equation can be converted to a system of n ﬁrst order linear equations. 3/9 2 Example 1 Third order linear equation: y ′′′ + 2y ′′ + 3y ′ + 4y = g(t). Deﬁne x1 = y, x2 = y ′ , x3 = y ′′ , then x′ = y ′ =⇒ x′ = x2 , 1 1 x′ = y ′′ =⇒ x′ = x3 , 2 2 x′ = y ′′′ = −4y − 3y ′ − 2y ′′ + g(t) 3 =⇒ x′ = −4x1 − 3x2 − 2x3 + g(t). 3 System of three ﬁrst order linear equations: x′ = x2 , 1 x′ = x3 , 2 x′ = −4x1 − 3x2 − 2x3 + g(t). 3 4/9 Theorem Linear system of n ﬁrst order equations: x′ = p11 (t)x1 + p12 (t)x2 + · · · + p1n (t)xn + g1 (t), 1 x′ = p21 (t)x1 + p22 (t)x2 + · · · + p2n (t)xn + g2 (t), 2 . . . ′ = p (t)x + p (t)x + · · · + p (t)x + g (t). xn n1 1 n2 2 nn n n Initial conditions: x1 (t0 )...
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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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