Question

Parts a, b, c, e, and h.

Place tranform X(s). 6, Solve the following initial value problems using Laplace transforms. da)) x5a = H(t- 2), x(0) = 1. bon al onoid li sin 2t, x(0) 0. b) x H 9lo c x" - x' -6x = 0, æ(0) = 2, x'(0) = -1 day sisini ds ovio astdo x(0) = 0, x'(0) = 1. d) r" - 2x' 2x = 0, x = e-', x" - 2x 2 a(0) = 0, x'(0) = 1. f) x" - x = 0 , 1, x' (0) = 0. x(0) e(0) 971 g) x" 0.4x + 2x 1 - H(t - 5), x(0) ai tdT bois h)x"+9x = sin 3t, x(0) 0, x'(0) = 0. = 0, x'(0) = 0. ) isd i) x" - 2x 1, x(0) = 1, ' (0) = 0. j) x2 H(t - 1), x(0) 0.

Answer 1

PN1 5 HIt-a) TL0)-1. OFG)-F(O FIG)- PN2 ol1l 9214 LT52 L[HIt-2) Tomalol H (applyng Laplate on bolth 6ido) Here LEPCH SFIS)- Plo) LIR]+5 LER] = L[HIt-2)] Coinat SFL9)-fro)+5 LE = L[HH-2)] G2492 Hore note P- LEPIJ SFI6)-Po) Here flo) = alo)=) LEP FI) JfojaoD Hee fit)-nF(S)-L Heavibide furehon mikn L[UIt-)J= e S SFOO)+FS) = F(5)= (SH)() 62+4 SFOG)-1+5 F(6) e chich 1s Yequdted 60 (0+5) FS)= 29 (0)2 lto) FtS) =G+ F(S)= L[aa6n = 0 _lappltng Laplaie on both oide 6 C6+5) GC6+5) L[q"]_ L[ q'] L] = 0 Hete E whith i6 Tequited 601? S2 FCS)-SPto)- f0) -[6F(S)-flo]- EF() -0 Here L[a] F(S) R(0) fio)=2 10)=D L[an L[Sin (2t)] applyig Laplaie hanoform on botholde) Fro)-612)(-1) -(SF(9)-2)-6F()- 0 L[] +LER = L[5net LLa'J+ LEn]= L[oInztJ 62 FC8)-26+-SF(8) +2-6F(S) = o

PN3 PN4 F(S)= F(O8-5-6] = 13-1-2 (1+3)(8226t2) whith 16 roqated 60ation FIS) = 2ら-9 62-9-6に-10 e+92- Gngt (o)=o (0)o cohich i6 reqtuired 60 L[ll+ga] = L[6instJ Daye HeLa+g Ln]- L[SInatJ 2l+2 t l0) o' to)= S FIS)-SFI0) flo) 9 FIS) 6219 LT 22l422= Lre'] (applying tarlare) Here ftolo) =0 L[")-2LJ+ n2 LE2]= L[etJ fIto)alo)o e Nny t F(O)=LE) LTe S- S Fts) S10)-0+9 FlS) Ube this. 62FCS) sfto)- Pco) nSF(0)-fo 2F(SJ= I (s+9) Flo) ら+1 S249 FI6) 0-(0)C0 fio)-lto) F(S) LEaJ (6249)2 (604) (649) cuhich i6 Yequalred s FIS) SIo) -26F()- o +2F(6) = 1+S SF(S) -0--26F(S) +2F(S)=し G2+2-26)F5) = 1+ 1+5
Here we 1tly applay laplace transform on both side of differential equations

Then Using formulas we solve this equations

And which formula use are mention here,