Question

PLEASE HELP solve ALL parts of this Diff Eq Problem in all steps clearly written. Thank you so much!

Answer each of the following: a. Compute the Wrornskian of the set {x, xIn x} b. Show that {x?, x? In x} form a fundamental solution set for xy" - 3xy' + 4y = 0 on the interval (0,00) and write the general solution

Answer 1

Answer à (a) wronskian of Set { 2, s?lan) wronskian = I x xenx alna 2x ex lnx + 2? à la 2u1nx ta E se? (222n x ta) - alnx-2x 2223lna +23-223 lux when a=0 : Wronskiam of = 23 when 2*0. (b) let y = x2 , Yz = a lux Differential earuation: wy" - 3x4'+4y=0 Now, y'=2x by," =2 in o we get. for y=J, Putting these value 282-38.22 + 4x2 = 623-622=0.

Thus, y=x? Solution of Satisfy 0.80, y, wb a the differential equation gonen by O. Yg = a lex Yį = 281nx ta y = 20.- +22nx tl =3+2lux For we y=Y2, putting these value in o get à (3+2 lnx) -32 (2x 2nx+x) + 4x2 lun = 3x² + 2x lux - 6x? lnx - 3x3+4x2lux Yq=2 lnx Satisfy o. So, yz is also a Solution of o Now, for all XE (0), wronskian of Ext, a lex} =&3 FO : {us, adax} is a linearly Independent set. Therefore, {x, ular} form a fundamental Solution set for 23 y^_347'+4y=0 on (0,00), General solution y(x) = C) x + ( x lnx where suc are arbitrary constant.