Question

1. Answer each of the following: a. Compute the Wrornskian of the set {e* cos 3x, e* sin 3x} b. Show that {e* cos 3x, e* sin 3x} form a fundamental solution set for y" - 2y' +10y = 0 on the interval (-00,-) and write the general solution

Answer 1

Soln 21 sin 321 3» @ We We h cos 321 The lanonskian is given by ere cos aze WE 24 sinares e 2 sin 321+BeļCUSB 21) e cog 37-38"sin (32) 2 e cos321 32 (en (eh sing2+ 3 e² 205(32)). "Sio (33) (e" cos (324–3 e "sinto) ru eze e cos(321) sio (32) + 3 è cos (31) su sin 2321) e 23 in 1321) cos(34) + ze eu 2 23e cos (321) +3e 2e sin 2,345 2u = 3 (cos² (321) + sin ? (32) 220 (i cose +sine=1) la = Be اک 6 claim sercus are easingh) "singh) is fundamental solution sel for 'y''-'+10.y=0 Le y = e le cos (34) y,' = en cos (32) ~ 3 e² sin (32) y," COS(BY) - 3 eu sincon) -3 easinow.tig ecog3W) g en COG (33) – 6 eºsin(33) 2 u e 11 : اولا Ling of ean becomeos

y"-2y'+loy = -8e%COS (323-62"sing32) -Recos bretoder sin (32) 1 + loere com (32) y'-ay'+wy,20 - Yizercos (32) is solution of y'l_ay'troy =o. eet Y2 = e re sin (34) Ya = e sin (321) + 3 e ze cos(3) 28 Gecos (321) . Y" 8 en sin.ua becomes LHS of egn 0 ya - mytroya = 682 (05(321)-80% in 3-2e2s in 1321) Cos(321) +10en sin (3) Ge 24 o 21 su: in=3e Irrgå toY2=0 =) Y2= sin (321) is solusion of eqn yl_ryltroyao since by poct @ (ie "+ to tHEIR) {ea (06 (32) , e sin(32)} it soha lineadly independ solution set of j'_ry'floy which is solution set. IN Zo fundamental solution