Question

6. Which pair of functions below cannot be a fundamental set of solutions? (8 Puan) 5e + 5, 2e + 2 e-21, te-21, 5 cost + 3 sint , 3 cost + sint e2%, cos 2x + sin 2x O1, x2, x2 In x

Answer 1

loet loen- loe - lock som fundamental set & let &, f, f...io are linearly independent function then it form fundamental set. if futz ifg...In are linearly dependent function then it can't be fundamental set of solr. Casell) W (41, 42, 9... In) to l. I functions Case li ) WL dugmy ys ym) 1.D functions in case one y,,G2, G ... In form fundamental of so 801". Case (ii) y,,G2,9 ... In can't be fundamental set of soft Set u In a set 5, 2e^+2 2 Wroksian w (y ,,Y2) w (5475, 22+2) = ya Seats 2e72 (5445) ze" sen 2 en 2e42) 5en ce tohle (al is not fundamental set of soph

•2t -2t te belezt te 5 ohraph b be -24 - 2t wly, 92, 93) te, 5 3 T! y 9. yg you 왤 where y paine is derivative -2t -2t -t - ze el 726 -2t e .(-2+) te ve(t-1 o. -2t -2t -2t l-2t. - ex ye x ye (t-1 Nolwe wohne (0) - te ( 4-1) 9 + gire ye?t 2tr e {1+2+) - 52-se"(t-1) - uele 1-24)? G -ut 53 - see - at -ut -ut -sett se -ue +8te UW Whoru -ut que to (b) is fundamental set of sol 1 Cost + sint, 3 cost tsint wly,,ya) w 1663t+3sint, 3607+Sint) y, Y2 y Y at t3 sint 3 (ostt sint -sint t3 cost sint test

(Cost+ zsint) (lost - 3sint) – (369€ +sint) 365+ - sint) { (ast) - (3 sin t}} - {(30st) - sint} (ght – asin²t - glost & sint (as?zt sin²t -91 ?zt sint). 1-9x1 (c) is Fundametal set of sole Come similarly we can check @ & @ d) en osantsinza TO - 8 to d een (asant sinan w (4,922 - 2 -sinane (2) + 2 692 an ce Cosant sinan en (2 cg2n- esinen) 2* 2 sinatie 2(aan, een 2 Cogan, ê 2 e27 et sinan (d) is not p. fundamental set of soph e 1, x², alogn w(1, x? x²logn) x² x Hogn axlogn + n ? // o 2 It zlogn+2 O an 2

1 (24)( 3+zlogn.lu)) Inlognan = 3 but unlogn - unloge - 2n an € 0 it is also fundamented set of solu Therefore b, c, e fundamental set of solu are