Question

onsider the differential equation y" - 7y + 12 y = 3 cos(3t). (a) Find r. 12. roots of the characteristic polynomial of the equation above. ri, r2 = 3,4 (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. Yi (t) = 0 (31) »2(t) = 0 (41) (c) Find a particular solution y, of the differential equation above. y,(t) =

Answer 1

Given that Y"_7y'+12y = 360$ (3+) How, let af solution of en be the Y"_ 74'4124 =0 Then, the auniliary equation is m²7m+12=0 - m²-ym-3m +12=0 = m(m-4)-3(m-4)=0 - 1m-4) (m-3)=0. = m = 4,3. (a) then, roots of chatacteristics Polynomial is 18,82 = 3,41 16 Then, cof of the equation is Ye=GY, + C2 Y2 , where Gg le are constant =l, e3t+Ceat [yi(t)= e 3t am (72 (t) = eht

() How let to find particular solution, Yp= A sin(t) + B Cos(3t) then DYp = A.3 Cos(3t) - B. 3 sin(3) = 3A COS (3+) –313 Sin (34) • D²Yp=-9 A sin (3+) - 9B Cos (37) And, DZYp - 7DYp + 12 Yp = 3 cos(at) - - 9 A sin(at)-9 B Cos(31) - 21 A Cost) +218 sin(t) + 12A + 12800S B4) = 3Cos(3+) + +218 Sin (34) +12 A Sint) in 13t) + 2-9B-21A +12 B} Cos3t = 3 coset 5) 209A +213 +129) sin(3 t) +4-08-21A +12 B7 Equating co-efficient we get -9A +213 +12A=0 and -9B-21A+RB20 = 3A+21B=0 =) A+7B=0 0 solving both, we multiply 7 with equation o then adding with 7A+49 B=0 -FA+B=1 50BT =) - 21 A+ 3 B=3 2) - FA+B=1 - B - 150

Then, A=-7B A=-17 / Hence Typ = - 17 / sin (3+) + 1 - 56 cos(3+)]