Question

6. Use the method of variation of parameters to find the general solution to the differential equation y" - 2y + y = x-le®

Answer 1

Given defferential equation is y" 2ylty=xtex. The deedereed equation is y"- 2y + y=o. The auxiliary equation m²_amt 1=0 =) (n-1) 20 m= 11 18 . The complementary function C.F = Ye = ciex tez xex. er, C2 are arbitowcey let y, (X) = ex, 12 (8) = xed, constants take the partientar integral as ype VICX) Y, (X) + V202) 42(X). where, Vix) = -Y268) R(x) de WOC) Yix) R(x) dx V2(X) > WOR) Where R(X) = xrex = ex

Now, W(X) = y (8) ฯ) (x) Y, '(X) ₂(x) keze 11 1 ex extreer e28 txean xe 2x e 28 +0. . Wic8) = -S x.ed, er at dre ezze - de =-X, and 42(X) = Sex dx. ed en X. e 28 s je dx = logge, :. The particular solution is ype - Red + X.ex logre .. The general solution is Y = Yet yp

+ or y = Gent cared- xex y = (1 + (2 X) ex & logg- 1) xer, sex logge or, y = (erteilex+ (logx-1) xex Thas the required solution is e, and C2 are arbitrary where constants.