Question

Need help with diff eq

Determine the general solution of the given differential equation (Show your work) 2x2y" – 4xy' +10y = 0 a) y = (x3 + c2x-1 b) y = (C1 + czln|x1)x3 c) y= claſicos ($71 In[xl) + cz|xpă sin ("ZI Inļxl) d) y = cz|x|3 cos ("7 In[xl) + cz|xl sin (977 1n\xl) e) y = cz\x{* cos(v11 In[xl) + czlxpă sin(V11 In[xl)

Answer 1

Given differential eguelion is 2x²y" -4xy't loyzo dy dy du = ēdy du da da= e du Idy = dy dh = 2=e kdy du a du da e Tă du ец dy) = dzy da? dlady da dy d ( + addre d (dy) du dx I dy + 1/4 dude = -1 dy tt is du Lue le udy du x2 du at 1 2 ( dày dy du 2 du (x²d²y day - dy 3 daz duz du Substitute ② 40 in 6, zelor de iau ce .4 (dy) troy =0 f 2021 duz ady _udy du du troy=0 = ) (4) 2d²y -ody duz du tlog Now the characteristic ean of above differential eqn is 2m2-6 mHOCO m²-3m+5=0 +359-4X 5 = M- 3+19-20 2 BEF = 3= si 2

e + , m, m2=miin Hence mia 3+Siti m2 = 3- ay the generel solution y (4) = Gemat ce ma te enacci Connaut (2 Sin an g 4 is given by / m,= m+ in me= min you) = ex 4 4 61 Tace) +6 Sim (5.2])-e*(9.6(84) + C, Sin (vic put x=e" an ezloge (6 con ( Fu In (iki)) = 2 gie) 948) log x=4, ale get +62 Sim (cu In(1x) Cie) |x2 ( 6 Gods. 1(1x) +G Sun (1001x1))) Where G&G are arbilary = 1x Const.
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