Question

4. Method of variable reduction makes use of one of the known solutions of a differential equation to find the other solution. Find the second solution of the given differential equation if one of the solutions is given. Show all steps. If yı:1) = et is one of the solutions, find the other solution of the differential equation using variable reduction. To do this, assume yz(2) = u(2)yı(2) = ue" and then solve the equation by substitution. (1 - 1)y" - ry' +y=0; 1>1. (4)

Answer 1

Given ditterential equation is (x-1yxyy=o one solution now & let 2(x) Solut u ex .y20x ue ue 2. y=uex +exu' + ule%+ e* u' 2 ue2 u'e u"ex Put above values in dierenHal eqn. x-1 y2 y +y=o (x-1) ex ut2u1+u' -xexCutu) ue = o (X-1 Cu+2u+u"] - xCu+u') +u=o fx-12 u 2x-2 -u' + cx-i)u-xu

tnis eqn Can Solve Now we bychange ot vasiable inro ducina nes variable Such chat vu" ditterential ean oecomes OS x-1 V x-2) V = 0 dv x-1) dx (2-2) V = 0 dv -x-2) Cx- dx Av -X-2) V Integsate both sides, we let. Inv= -1-1- (x-1) dx -LS-)d nv 1nv -Cx 1n x-1] +C n(X-)- +C, InvE

(InCx-)-X [Kconstant o interah V= ke In(x-1) V ke V- ке х Сх-7 VkEe nous 1 u-Ke e xe*]+c uKx eC hous we can choose constanbs to be anything, to clea out Constants all extraneous

nouslet K 1 u -x e" X = -x exe uex 20x)=-X solution tdY ditferenHaf equation is wyitten as A YI () 2 Y2 ( AE Bx