Question

Find the most general solution to the differential equation below (give an explicit final answer in the form “y = ... "). y"' + 5y' + 6y = 24t + sin(t)

Answer 1

be the y"+ 5y'+6y= 24+ + sin(t) differential eacuation, of Second order non-homogeneous. solution of homogeneous part of the given differential equation. y = amemz and y= Am²emt. Let trial y= Aemt be then reprace yoy yo on y"+59'+6420, then get the causiliary equation, m2 +5m+6=0 m² 3m +2m +620 or m+3m +2) =0 m=-3, and m=-2. 637 Bezt A,B therefore the complementary solution solution is y(t) = A et Be are arbitrary constant, Now we calculatlate particular integral.

het uplt) = AtBt + esinet to cos(t) by method of undetermined co-efficient then, Ypc) = B te cosct) - D sin(t) and ya (t) = - e sin(t) -D COS (+) replace up. ye and he y + 5 hp + 6yp = 29z+sin(t) -esinato 5D sin(t) DOOS(+) + 50+ be cos (t) ŁAt+ sin(t) + 6A+GBt + be sin(z) + 60 cost = 5 e sin(t) +50 eos (t) +5e cos (t) - 50 sinct) + 6 A+58 +6BZ = 24+ + sin(t) Sin (2) [58-60] + los (t) [50+5e] + 6 Btt GA +50 - 24 t + Sin (+) and compare from both side coefficient of sint, cost t we poneer of get A = - 2 GA +50=0 GB- 24 > B = 4, then 20. and 50+ be . and 50-5DCI - 5D = -56 50-5D=1 5C=1+5D 5 e = t-te toez1 to. to and 20 therefore Yp (t) = + at cos (+) Sen (t), 10 10 the hence general solution is yeta uut y plt) 3t y=Aé +Bezt 20 +44 +1 Eos (+) io Sinct) arbitrary constant, solution, А are be the general