Question

Consider the differential equation e24 y" – 4y +4y= t> 0. t2 (a) Find T1, T2, roots of the characteristic polynomial of the equation above. 11,12 M (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. yı(t) M y2(t) = M (C) Find the Wronskian of the fundamental solutions you found in part (b). W(t) M (d) Use the fundamental solutions you found in (b) to find functions ui and Usuch that a particular solution to the nonhomogeneous equation above is given by Yp = u Yi + Uz y2. м U1 = м U2 =

Answer 1

o We have (D² - 4D+4) y = e²+ 1 +² - 0 a) Put D= m in honwgenious port of - m-4m+4=0 gives (M-2)=0 m=2, 2. Thus soots are 2,- 2.72-2

b) Thus, the cof will be (F= (6, + C₂ *) e ²t = C, e24 + (te²t : Fundamental solutions will be -) Y,!!)= e²t 92/8)=4e2t c) W11) y Y, Y y, 2t e at tet e²4 +2te²t 4t 40 4t = etateate at -atett d) Particular solution using Woonskian iş given by formula 4p(x)= -4, (*) (yz(x)faldt у Wly,:42) +4₂(X) J :(*)f(x) dx Wly,342) Here U,(x)=-| x2.e4x 2x 2x хе ê dx - logx+C x 40 2x Uz(x)= Semedy x².e4x : +4 F: f(t) = é* /t? i-e. f(x): e. f(x)=é*/x+] "Thus, u,(t) = -logt + c 12()=-11t+C.