Sol.4.4

# Sol.4.4 - 4.4 Exercises in ecluation(4.2 is r'2 1 0 which...

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Exercises 4.4 - . + anxiiiary ecluation for the clifefrential ecluation in (4.2) is r'2 + 1 : 0, which has '-', irrragittary roots r: t'r. Therefore, a general solution is given b;' Y(t):c1cosl*czsinl' non,flncl corrstants c1 and cz so that g(t) satisfies the initial concliticins irr (4.2). sinr: g(0) : cr , cos.tr : E'(0) : (-cl sirr t -l r:zcosl) l,-o: cz. -.r(-.fore) U(t) : sin r cos f * cos r sitrl . Theorern 1, Section 4.2, the solution to the initial value problem (a.2) is urriclue' rs. u(f found in part (b) must be /(t) meauing that sin(z * f) : sinrcosf * cos:rsinl. -:ES 4.4: Nonhonrogeneous Equations: The Method of Undetermined Coefficients -l,ocl of undetermined coefficients carrbe used to find a particular solution in the , n-ith Trr : 3,rl : 0, ancl p : 4. sin r * 'r.irt the right-hand side is of the ,-!(rrrrogeneous terrn sirnplifies to -,4. r :e'"Sillu, fortn, for which (15) a,pplies f (r) :4:r siu2 :r * 4r cos2 r : 4:r (sin2 r * cos2 ,) : 4l ' :,.. t|e nrethod of tindeterrnined coefiicients can be used, artd a particula,r solutioti -, nn (14) with rrz: 1 ancl r' : 0. .-: {-,alr use the method of undeterrnined coefficients beca,use the right-harld side of ' r. eclra,tion is exactly of the fortn, for which (15) applies. : 0 is not a root of the auxilial)r eeutrtiott, t'2 + 2r - I:0, wc choose s : 0 in i s:ek a particular solution of tlie forrn 37o(f) = Ar:. Substitution into the original -, r'iclcls (Ao)" +2(Ao)'-.40: 1o Ao : -10. 119

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Chapter 4 Tltlts, Ar(t) = -10 is a particul:rr solution to tlie given rronhomogeneolls equatiol. 12. The a,ssociaterl auxiliary equation, 2r -t r: 0, has the root ,t' : - 112 10. So, w-e t. s:0 in (1a) ancl look fol a particular solution to the nonhomogeleolls eq1ati91 of -r fcrrrn t:r,(t):A2f+Art+Ao substitution into the original clifferential ecluafion yielcls 2:r'r(t) + :rr(t) : 2 (2Azt 1 A,) + Artz + Art+ Ao : Azt2 -t (4Az + A) t+ (2A, + Ao) :,,- By equating
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