2130_Chapter7_Notes

x pn1 tx1 pn2 tx2 pnn txn gn t n if

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 )...
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