Sect7.4 - <[Z\ .y' are gold—toms a; 'i u .14) W:...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: <[Z\ .y' are gold—toms a; 'i u .14) W: Wfaflsbiam oi; ’1 and ‘K INTRODUCNON 70 Dawn mm. 50w 710m 0) \ - d7“ (2) ‘m 61151“ m C”? dx‘” ‘ -—-—’ Yb " ‘X -—-.' : 4' X‘ 'f" “ ._.l’— Kl“ at 3 at] dt dt ‘ / m (t) 1. d1 . dag: 1} 78m Xx”) ‘ , a = det + det an? ant,“ U} _(2) > f —/ X all: At L . v a- r _ (31:1 putt» pzm we) - ' ~ "/- '. ® Ejtsy 1 ~ 9&1 => de\u\ i7.‘{p.1! Subsmwe Ww WWSScms {30W 7t amd ——. LVL+D w M;th 96 if” i‘ft) " def m m (2) (z) k?2‘ \ P211; Pz‘yw + {32sz .. (0 (t) {H , .(t‘fi __ , (z -n (2) (1) - FHX’ X + at,in X1 Play )1; PIE Z 7'2 m 3, .m m (z) m (z) {p P011“) 1‘ + $21,117:: 1‘ - P3111 1) 'Pze, ’12 11 O 7 () 70 I ) ( .. (I) ( ) (z (I z , I l) / N____ ‘‘‘‘ _4 H.-- _, W ('0 (z) m m (a l m m (a) \- Pu [1‘ I, f— 1, "LI 1 k PZZ[%Z)I‘ 7L; 1‘] / v» I 1451‘ _ ................ 31%;“ 2:er .wxmmgvmmwmfimw .......... -M \\ M _ R[Pu[t\v 1- P7 (a \d{: J! W J ‘ J .- .30 m ew‘ vmcgt. vafiw ob t, 36 C-OJ Wham W £3 L3 Section 7.4, Problem 4 _ V 1")“: 1‘: :‘SL'XK‘: frl‘Yl‘i" 52:13:33 T:"‘?;§?i£f.‘i‘i‘i"f’ iYi‘m’ q(t)y = 0 corresponds to the system: I 221 = 0101 +322 = ‘Q(t)$1*P(t)$2 Show that if 56(1) and 33(2) are a fundamental set of solutions of the system of equations, and if 31(1) and ye) are a fundamental set of solutions of the second order equation, then W[y(1), y(2)] = cW[a'c'(1),f(2)], Where c is a nonzero constant. Hint. y“). and ym must be linear combinations of x11(t) and x12 Solution. Efficiently answering this question relies .on a result from problems 2 and 3 in this section. If one has worked it out, all is well; if not, one might use} the result of that problem without proving it. Working from the results of problem 2, one may note that the quantity P11 + 1222 = ——p(t) (using the notion in Equation (1) of this section). Therefore by the results of problem 3, W[a'c’(1),f(2)] = ce” “(t)”. For the second order linear differential equation, the Wronskian, using the definition of it introduced in Chapter 3, must satisfy the differential equation W’ + p(t)W = 0, and so therefore W[y(1), y(2)] = cle_fp(tldt. So the two values for the Wronskian, ce‘quam and cle‘ hm“, must be con: stant multiples of one another. An alternative but equally valid proof, albeit one depending on more algebra, is to make use of the hint that ya) and 31(2) must be linear combinations of 5(1) and 5(2)- SO, ya) = 0111m11 + 06129612, and 21(2) = 01215511 + 01223312. Calculating the Wronskian directly, one gets: WW0) gm] 0113311 + 06125512 0211011 + 05221712 I I I I 011111311 + 012$” 06215311 + 01229312 I _ i I (011022 —' 012a21)$11$12 — (01110122 " 06120521)$12$11 (01110.92 — 01120121)$11$22 — (0110522 — 0120121)$12$21 That last, slick, line, makes use of the fact that w’l = 9:2 (by assumption, in 2:“: .: ’ WWW“ V _ _ _ t , , , , , ,W WWW wwwaW; .‘-w..m_fl..mmw.wm_mwwtwm. W. ."V web-m»mew«mewgzfigmgwhr,gfiwwmwwW Therefore, W[y(1),y(2)] = (04110122 — 01120121)W[y(1)’y(2)] = eta/[5(1): 13(2)] Or, as hoped, the Wronskian of ya) and ya) is a constant multiple of the Wronskian of if“) and 5(2). [5) Show what We ffnered Scmh‘cn 052' ‘2’; PM? + 374:) is the Swm ‘ POW 0w ScLuL'Za/x if?) of) awath (1rd W nay-bend swmx gi(c) LQ‘E $59) he an 03ng Semen 76’: PGBY (-37%): / _ ” -( ‘) z ‘ ' The“ @033 ' Y P SCIMS \Hhe \ncwoSé/neo'ws eaguccucm. ’2 = P691. _ ‘- _ ) -m -> (Alfiv 700‘: CI.“ " 02’?!“ o— + 6 7M“) 45 => Gilt) - 77W * 6’, 77m * * 9:77“) 2 70"” c 2 CU'LZ .de'andLnt .41» (La mkrtuals mot Mogadqu Zero_ © A’c hast one obiune Caeéékm [331(k) u} we, sgsw'mst be, (14% Con+wu¢ug 'af- > 75:0 @ The Que ed Q ' ' . —— 8 r vdbd’CCfl 06 we 333% 7:31. I = cC7501 + Cziay he.) 1- 1‘ a Ff n {a 1L "‘ = L‘ a '* ‘1 ’ L zj Zt Then J— t vi 7, ‘ I. X L 1 l o 'Dh‘de (3* (cu: ‘ -‘L I ‘3 it ‘ t4 0 add +0 ‘ 42% I H‘ O emote 7*“ f ' __.->7 (Ow ‘0 — ‘ .L b3 "5’6 y ‘ ét ' H O 0 2t ’25" \ ‘ I E a ( l x \ O t “2-51 2t" L '\ MuH. Zfld (-0“) - L bj ZtJO‘dd‘w ‘9‘. (w) 1 o ‘ ...
View Full Document

Page1 / 8

Sect7.4 - &amp;lt;[Z\ .y' are gold—toms a; 'i u .14) W:...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online