Question

Find the particular solution (write the answer of each stop) of the following differential equation given the two linearly independent solution of the homogeneous equation Y-X.Yaxin(x) zpont - 2x + 2y=xn°%.

Answer 1

We solve this differential equation using the method of Variation of parameters

Soletion :- The given differential equation is : 2x² dy - 2x dy , sy= (1) doce doc which is Ereler- cauchy and order differentical equation. mcm-1) a enex) Ret se dhe te m 0 a dy dic The characttistic eqreation is: a mom-10 2m + 2 = 0 mom-1) - mt 1 = me_ - m - m + 1 = O me-em + 1 m-ije m= 1, 1 which are real double roots. i. The complementary function Yc is: Ye = (ait ce ln ac) act qxt (ex en ac The two linearly independent solutions are & (which are already given) Y = ac Ye= x en a Now, we tind pearticular Solution of equo first diride equation (1) by ex, we get - 3 - 2 de la + 1 e Y = x lncus en cao is let rex) - eneca) dice 2x2 2x 2x Here, we to find use the method of variation of parameters its portioner solution.



We x -", W= 2 la enxcss IL find wronskians, W,w, & W2. Ye alna ye 1 x. 1 t en aci) x In ac 1 tena X C1 + en x) - xlna = x + x fnx - xx xx 1 OC -x In ac Wi= 1 i 4 Yz Y2 x en oc 1 t enac 101 111 121 W(2) ental data X 2x 200 19 Waz 1 al = 12 a 1 = 0 Using the method of reviction of parameters! W2OC) rcx) dic Yp (x) = Y, C f W (x) r(x) dx + YG CK) f al pena a encx) dx + x lnx =-2 / 1 linx) - 2 dx + x lnx f docx) . u I dactic Using the sule: Parayi plet) dx = Cf009+'+o -2 x encx) en 3 (0) + c Lan Ca + - ente) + s dn 4(x) + c Yocx) = x em coe) [+] 24 nt en 9 (L) 4 + 2 ac en'ex) + c mela nn