midterm_2_review (dragged) 5

Midterm_2_review - 6 MA 36600 MIDTERM#2 REVIEW Chapter 4 §4.1 General Theory of nth Order Equations • An nth order linear differential equation

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: 6 MA 36600 MIDTERM #2 REVIEW Chapter 4 §4.1: General Theory of nth Order Equations. • An nth order linear differential equation is an equation of the form dn y dn−1 y dy + P1 (t) n−1 + · · · + Pn−1 (t) + Pn (t) y = G(t). dtn dt dt We wish to solve the initial value problem P0 (t) n ￿ i=1 • The sum Pn−j (t) y (j ) = G(t) where (j −1) y (j −1) (t0 ) = y0 , i = 1, 2, . . . , n. dn y dn−1 y dy + P1 (t) n−1 + · · · + Pn−1 (t) + Pn (t) y n dt dt dt is a linear ￿ operator. That is, given functions f = f (t) and g = g (t) as well as constants c1 and c2 , ￿ we have L c1 f + c2 g = c1 L[f ] + c2 L[g ]. In particular, if {y1 , y2 , . . . , ym } is a set of functions satisfying L[yi ] = 0, then y (t) = c1 y1 (t) + c2 y2 (t) + · · · + cm ym (t) also satisfies L[y ] = 0. • Say that we can find (n + 1) functions y1 = y1 (t), . . . , yn (t) and Y = Y (t) such that each yi = yi (t) ￿n (j ) is a solution to the homogeneous equation j =0 Pn−j (t) yi = 0; the Wronskian is nonzero: ￿ ￿ ￿ y1 (t) y2 (t) ··· yn (t) ￿ ￿ ￿ ￿ (1) ￿ (1) (1) ￿ y (t) y2 (t) ··· yn (t) ￿ ￿1 ￿ ￿ ￿ ￿; W y1 , y2 , . . . , yn (t) = ￿ ￿ ￿ . . . .. . . . ￿ ￿ . . . . ￿ ￿ ￿ (n−1) ￿ (n−1) (n−1) ￿y (t) y (t) · · · yn (t)￿ L[y ] = P0 (t) 1 2 ￿n and Y = Y (t) is a solution to the nonhomogeneous equation j =0 Pn−j (t) Y (j ) = G(t). Then there exist constants ci such that ￿n ￿ n ￿ ￿ Pn−j (t) y (j ) = G(t) =⇒ y (t) = ci yi (t) + Y (t). j =0 i=1 We call {y1 , y2 , . . . , yn } a fundamental set of solutions. • Say that {y1 , y2 , . . . , yn } is a set of functions satisfying the homogeneous differential equation dn y dn−1 y dy + P1 (t) n−1 + · · · + Pn−1 (t) + Pn (t) y = 0. n dt dt dt Abel’s Theorem states that there exists a constant C such that ￿ ￿t ￿ ￿ ￿ P1 (τ ) W y1 , y2 , . . . , yn (t) = C exp − dτ . P0 (τ ) P0 (t) • A collection of functions {f1 , f2 , . . . , fn } is a linearly independent set if the only solution to the equation k1 f1 (t) + k2 f2 (t) + · · · + kn fn (t) = 0 for all t is k1 = k2 = · · · = kn = 0. The following are equivalent: ￿ ￿ i. W ￿f1 , f2 , . . . , fn ￿(t0 ) ￿= 0 for some t0 . ii. W f1 , f2 , . . . , fn (t) ￿= 0 for all t. iii. {f1 , f2 , . . . , fn } is a linearly independent set. §4.2: Homogeneous Equations with Constant Coefficients. • The function y (t) = ert is a solution to the constant coefficient homogeneous equation dn y dn−1 y dy + a1 n−1 + · · · + an−1 + an y = 0 dtn dt dt if and only if r is a root of the characteristic polynomial Z (r) = a0 rn + a1 rn−1 + · · · + an−1 r + an . The equation Z (r) = 0 is called the characteristic equation. a0 ...
View Full Document

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.

Ask a homework question - tutors are online