Question

Answer all of the following:

Use Laplace transform to solve the following ODEs (show reasonable detail) j(t) + y(t) + 3y(t) = 0; y(0) = 1, y(0) = 2 (t) + (t) =t; (0) = 1, g(0) = -1

Answer 1

Some lell known fermula h[eat Sinwt] e+g) ²+wZ L (e-atconwt) = O+9)2+wz L(E) =L(t)=Yz.

ECf C+) = eA Preselt- hCf'(H) = s hlfe)– fro) h (f"(t) = hlf"(t) 00 A?Lifes) -afro) -f'ro) kGr) = A"Lifeu) - fro) -r"fio) --- - Pr%0) of %3D 9 d"Ct)+yu4) +3y4) = 0, yro)=!, gl0)=2 y"lt) + yt) =+ y co) = I, ď'ro)=-) Let's slaut with pant c@). g) +y'Lt) +3y\E) =0 Appy Laplace hromform both the aide g"ct) +ylct)+ zYt) = hl0) = 0 h Cyucts) + use formula ® in above equation Since Loplace Hy'Cts)+3hly(ts) = 0 (Fronform satinfy braity popng > 6=LCyct)- fr0)-io)+A Hytet) - ro) + 3lye) = 0 (A+ p+3) Llyıt)- A.L-2-1=0 ) (A2+p+3) Llyet) = A+3 %3D st3 Llylts) = B2+A+3

L(yets) =_a+3 and s+3 ニ s2+p+ 3 sa+2(^/s)+3+%¼-V4 иCт pls Dt 3 (A+2P+ Y4)+(3%;) (Aty))e+(") (o+)+(3%) (A+%) S. 2Cyin)) = (>+/) (5/2) (p*x) *+( (e*)=+( +(ツ) (8+%)²+ 2. y) = L" (6+%) }+L"{ 5/2 (9G) ことt e Cop 4/2 Sim ()+ ( %3D

msn-f%3A16425681343150898548simpl=msg-f3A16425681343 t)+ Sin(Sit (© yut) +ylt) =t, ylo)=, yl0)=-| apply Laplace frommferm both the mide h (ynie) + yee) = LynCt) + wy(t) = Yz sE Liyt))-> yeo)-y'lo)+ Llylt)=' Lyit)) (x+)-sci)+|=¼2Liyet) +(1)= L(t) cs+)

https://mail.google.com/mail/u/1?ik=8e447e5d14&view=pt&search=all&permms (1+x2) Leget) = #-(1») >L'yrt)) =atrioany tro (I-) Liyit) = | +12 yet) = L"()-2"6)* Lf y t) = L'(+ %3D +A2 2 Sint + 6nt (1)R