Question

Find the general solution of the following non-homogeneous differential equation d 2 y dt2 + 2 dy dt + y = sin (2t). (2) Now, let y(t) be the general solution you find, when happen if we take lim t→+∞ y(t)?

2. Find the general solution of the following non-homogeneous differential equation dy dy sin (2t) (2) 2 +y= dt dt2 Now, let y(t) be the general solution you find, when happen if we take lim y(t)? t-++oo

Answer 1

And differential Given equation non- homogeneous is 2 + 2 dy + y = sin (24) - (6² +2D+1) = sin (21) where Deut Auxiliary equation will be m2 +2m +1 -0 (m + 1)2 = 0 m=-1,-1 Complementary function (C.F.) = (C, + (GA) e + : Now particular integral (p. I.) = 1 (02 +20 41 Sin (24) - Sin (24) -2² + 20+1 = 1 - 4 + 2D + 1 Sin (2+) (20-3) Sin (24) multiplying by (2D+3) in the Numerator and Denominator

(p. 1.) = (2D+3) - (20+3)(20-3) Sin (24) (2D+3) (4D²-9) Sin (24) (2D+3) Sin (24) -4.2² -9 (2D+3) Sin (24) -16-9 (2D+3) Sin (24) = (PI) = [4 Cf(24) +3 tin (24)] bg саще Dad dt - General solution will C.f. +P.I. be y (t) = y (t) = (C, + (24) et (Acos (24) + 3 Sin (24) 25 Now we know that as to, then - Cos(2A) and sin (24) does not exist. Hence when t t , then solution of given non- homogeneous differential equation does not exist.