Sol.4.5

# Sol.4.5 - Exercises4.5 pat'ticular solutionof the torn...

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Exercises 4.5 pat'ticular solution of the torn yr(t) : t,(Aocosf * Besinf), ancl choose ' is not a root of the auxiliary equation, 14 _ gr2 _ g _ 0. Hence, Nqrss\ \ B\sN\ fl'r,(t) : Bo cos I - ,46 sin f A;(t) : -As cos t - E.11 sint a;' (t) - - Blcos f * ,46 sin f r|n) (0: .4s cos I * Bo sin f . - 8Ap - -4Aocos I - 4Bo sin I : sin f 1 :0, Bo: -) =+ ao1) : -sinf. 4 oyr-t 4 :lES 1.5: The superposition principle and Uncretermined coefficients Revisited - :: cos2t and, Sz(t):: f. Then At(t): (1/a)sin2f is a soluti.n to a"+2a'*4ll:g{t) : t l4 - ll9 is a solution to ,J" + 2U' * 4y : gr(t). ' :iqlrt-hand side of the given eqnation eq,als gz(t) + a(t). Therefore, the -'' rrt u(t) : az(t) + n(t) : tl4 - 1/8 + (1/a) sin zt is asolution to the equation -)l'+4!J:t*cos2l. rii €Xpr€ss 2t -3cos2f :2g2(t) - 3g,,(r). So, by t'e superposition principle, -'.ired solution is y(t) : 2a2(t) - Jur(t) : tl2 - rl1_ (Jl4) sin2t. . 11f - 12cos 2t :7lg2(t) _ I2gr(t), the function a\t) : rrs2(t) _ r2y1(t) : I7tl1 _ lrlS_ 3 sin2f - ltrtion to the given equation. \j\r\u ) -- vP -svi =+ Ao : i ri

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Cliapter.l 4. The corresponcling homogelteous ecluation, ,y,, * !J, : 0, |a,s tlie assgcia,tecl auxili.- equation 12 + r' : r(r" * 1) : g. This gives ?, : 0, _1, alcl :r gc:1era,l solrrtiol to tt lromogeneotts equatio tt is y1'(t) : cr * c'2,€-t . cl0nrbining this sohltion with tht, lrarticlil- solut'ion, ap(t) : r, we fi'cl that a gene'.I scirntion is given l,rv Y\t) : ap(t) + an(t) : t t ct I c2e-t . 6. Tlre corresponding auxiliary equation, r.,z * 5r + 6 : 0, has the roots r, : _3. __ Therefore' a gerieral solution to the corresponcling hornogeneous erluation las t6e f, - an?;) : cle-2''+cze-3'' By the superposition principle, zr general solution to 6re origir nonhomogeneous equation is !l:r) : ar(r) + gn(r) : e, + 12 I c1e-2, I c.2e-:J,, 8. First, we rewrite the equation in slattrlatrl folrrr. tlral is, 'y" -2y:2taullr. Tlte corresponding homogeneous equa,tio D, U,, _ 2!l :0, ha,s the associater-l auxill ecluation t'2 - 2 : 0' Tl.rs r : t1/1, anrl a, gereral sohrtirn to the Lornoge. ., erlrral iorr is lln\r) : (\ertr * c2e-fr,. Corrrbining this with tlie particular solutior t, yr(:r): tan r., is given b1' tl(r) : yoQ:) * an(:r): ta,1 t: -f crefr" * c2e-fr, . \O. \\fe ca* wr\t,e \he *onholrloBeneo.s term as a c\iffere*ce ("t + t)' : e2t * 2tet + t2 : gt(t)+ 92(f) + 93(f). Tlre functions p1(t), gr(t), ancl 93(f) have a frrrrn suitable
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## This note was uploaded on 03/30/2009 for the course MA diff eq taught by Professor Fenn during the Spring '09 term at N.C. State.

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Sol.4.5 - Exercises4.5 pat'ticular solutionof the torn...

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