Question

(1 point) Solve y" + 2y + 2y = 4te-t cos(t). 1) Solve the homogeneous part: y' + 2y + 2y = 0 for Yh, using a real basis. Note the coded answer is ordered. If your basis is correct and your answer is not accepted, try again with the other ordering. Yn = C1 e^(-t)sin(t) +C2 e^(-t)cos(t) . 2) Compute the particular solution yp via complexifying the differential equation: Note that the forcing e * cos(t) = Re(el 1+i)t). You will solve the complexified differential equation y" + 2y + 2y = 4te 1+1). The natural frequencies are: -1+i -1+i and -1-1 (Note: the ansi (Note: the answers are ordered. Try a different order if your answers are not accepted.) The forcing frequency is -1+i -1+i Yp=t(e^((-1+i)t) (At+B) (use A, B etc, as your undetermined constants.) To solve for the undetermined constant(s), impose the criteria that yp solves the complexified non-homogeneous differential equation: y + 2y + 2y = A exp( tt Note above: you collect like terms in t. The coefficients are in general complex. So, A=( B= Plug that back into y, and extract the real part for the original differential equation we set out to solve: Re(yp) = 1 Your full solution is y = yn + Re(yp).

Answer 1

you + xy' + ay = até + corte) -0 () Consider the homogeneous equation y" + ey' + 2y =0 Characteristic (auxiliary) equation is m'+3m +2= 0 => m=-41 V 24-48182 2X1 :: m = -2° V4-8 = -2 + 1-4 =-2 2i = -1 + i Rule for characteristic roots atiß , then Yn = ent (a cos(pt) + Cq sin (st)) Here, &: iß = -1 ti cs Sana ametot (c. conlu ) + G ain (1 t)) CamScanner

:= (, costo + C, diver) y G c* coste +ge tainty (2) Comides et tilt , tr ( cao (1) tủ 4 (1) E et conlt) ti ét ain(t) :. et cost) = Re (ettist) Now we will solve the complesified differential equation y" + 2y' + 2y = 4techt); The natural frequencies are Hi and t-i The forcing frequency is alti Ho = ť (At+) -lit Here -ti is characteristic rost with multiplicity 1 ...S=1 :: * = t (At +B)elit Scanned with CamScanner

% = 16€**}* ) (A++8) = (A4° +84)861+65+ Yo' = (24+ +6) EL**)* + (at +62) 2*) (1-1) = (24+8) <++i) +(-At? - 8t + Ait? + Bit)E" = ([**]( 244 +- ++ - 8t + nitº + bit) Fellt)t ( (24-6+81)£ + (A1–ADŁ° +) sy" - €646)! [(94–6 + bc) + 2 (ai–A}++ o] + 486)+ (1+0) [(94-2+ Bidt (n. -Ayt° +6] = e*)# [ (8+ 6c) + (941-)+] +41+)+ [(-2478-6i +24i- Bi + Bit)t + (-Ai + A +A32-ai)t' +81 – 87 =&ti)| (-2Ai+A-A)/2 + (246-24 -2A+B-Bi+qai Cs Scanned with -Bi-B)+.+ (2A– B+ Bi + Bi – B) CamScanner

= elitist (-dac)t? + (+-4A – 366)+ +(29 -26+81) Coneider Yo + 2y/' + og = CM2)t | (+264) £® + (4 10 –44-20i)t + (24-28+381)] +del+te) | (–a)t? +(24-8+6+6] + delt{ pe + 86] = 264+0)4 [(-241 + 2Ai – 24 +24) ++ (4A1 – 4 A-2B1 +4 A-28+2 Bi +26)t + (2A-264 261 +26)] = 66-) [0x' + 4ait + (84+26)] - Ael**** [0t+ 4i + +(2+241) wowe, Yo is particular solution of y" + 2y' + 2y = 42 C-list Scanned with CamScanner

:. " + 24 + 37p = 4+el+) → Rel++)! ( fi ++ (av 341) - 4tectie 2A(46+ + (444)) - 4+ => 41A = 4 and 1(2+ 381)=0 - Ai =L =) Art ** = -i and Afd+ 25i) = 0 =) 2+ 38 i=0 27 2-28 = 0 => 1-B = 0 colt => B = 1 cs Scanned with CamScanner

.. A = B = 1 % = + (&?*0)%) (+it+2) + + -L-it'tt) ** (corte) t i din te) =€* (-it' +1) co lt) + il-it'tt) e *rint) =+* [-it".0t + tent - it'eint tit simne] =6* ( cost + t'aimet + 3 (trint +*cat] . Relys) = ét (t wont + e' sint) =tet (cort + t sint) : Full solution is yo tu + Re (yo) i da cause it in lt) + G &* cost +te(cost + taint) CamScanner