Question

9. Find the general solution y(t) using the method of undetermined coefficients.

Answer 1

volution: +0 Given equation is y"+ 4y'+ 4y = x 22+ Homogenous equation of is y"+ 4y'+ 4y =0 -- is characteristic equation of x+48 +4 -0 (x+2²=0 d=-2,-2 The roots are real and equal Basic solution of equation © are ya e-2t Y = teht General solution of equation is Yo = 99, +Czym . = 4e-2t + te-2 Yn = [(+9+] ezt → Now from equation Flutet By the method of undetermined coefficients particular solution 4 = (Ax?+Bitc)2X $(t) = (4ť?+BE+C) e 2+ y' (t) = 2 ( 48°+BE+C)e£+ +(2.4+B) ezt

=[20k++ (2 6+ 2A) + 24 +5]C.PL Now y"(t) = 2 [20&+ +(264 2A)+ + 2C+57e44+ [+A++ 284 2A]e 25 = [4x4 + (4818A)t + 4C + 40+24]e 2t Then y"+ 4y' +44 - telt [a At++ [48+s4]t +46+48 +24]e+4 [24tºt[18+24]+2C+8%e**** +4[ne®+6+ +c] ezt a thout 4 At + 4bt +SAE + 46+43 +2A + SAC +8BE+SAL +86 +48 +4A 4BE + 4C = ť BAL? + 4At + (2Bt+4Bt + 16AC +12C+4648B +9AK 16AL? +1684 + 16AL +16C +8B+ZAK 16 AE =k A = te , 60,C=0 then 4 = [4+*+B6+c]elt = 12 e 2t General solution 4(+) = Y, (t) + 4p [4) = ce-2t +ę te-2t + I the 2t