Question

8. Solve the initial value problem using the Laplace transform method x" + 2x' + x = t + 8(t - 2) x(0) = 0, x'(0) = 1

Answer 1

Given ve is all tza tx=ttsct-2) V) Taking caplace transform, we get, LC3] +2.4[341+ [x]=14t) +48(4-2)] & prst-pecal-alo) +2 [pa- reco} +2=2 te 1:42] =-alo L[Dell] --p1a-810 Dee) LEEhsan monto Lsct-ayjce of 2- LX Here ap² să-+zpłtă ad=a, na 2 tezer te2p 1 ener Phulle (6²tapta it te zo (ptuje zł alt for testo +1 6+)2 b2cftiga L[] 13132= pe (htige (b+y find a, we shall take inverse Laplace transform shall first resare into partial fractions. p² (pt); Now, 1 +)-r p² (pt) pe chall q RHs, for this 1 bb11 b' ) P-1 Þ' (PV) pobt

+ - toz b) plet) +*+12 2 bert) a 2 [C6+1-6] 11 poto Pubti) IP (Pty bu 276 + 1 to (btil? Setting in (2), we L2121t (6+1² 2p + (pty2 poty? احاد 2 p² 2 pti b+y? +4 (ht112 obec [Hb+20 ('[bbt22 [A] -26"[+1+2011strial + I"LE bikat at CA I'll at El Pon) "[ht =='fhta) C'E=1 (ta) = at home [btiel - et illa Els(etaje I (SW)] r. Ellalat 116 E' (BWP)]=FV), then [/ Teller spil Fua Putting Ht-a) at Len - det =(-2)2(t-3) -2p ac HC-2) above volna in (3),

x=tthet_2tztett (t- ze(t 2) Hct-2) which is required solution,