Question

6) a) Solve the following differential equation using the Laplace transform method. dy = 1.87ylt) + 4.05 y0) = 1 You may need the expression, 1.05 4.05 s(s - 1.87) 1.87(s - 1.87) 4.05 1.87s [8 marks] b) Solve the following differential equation using the Laplace transform method. dºy + 2.61X + 6.55y(t) = 0 y(0) = 1, y'(0) = 1 2. You may need the expression, s +1 +2.61 52 +2.615 +6.55 *2.01.2015 - | 1+2,61 (8+2.01) + ((6.55- (2,01)2)) (+2,01)*+((6.55 – 2.01)2)) +2,02

Answer 1

619 = df = 1.87 Y(u) + 4.05 Or, yht) = 1.87 y(d) + 4.051 Taking haplace Transform on both side fycts} = 1.87444641} + hp 4.05} 07, 6846) - Y(0) = 1.87 461) + 4.05. Or, syks) - 1 = 1.87 Y(A) + + 05 [: Y(0)=1] ox, Y(A) (A-1-87) = 1 + 4.035 Or, 40) = 184 ov, 46) = 4187 + Toon 0-41545] ONS MUSICA L Taking mverse laplace transform on both side y@="64409} = 144-1.87 34 [164 337 y (4) = 1874 4:05_e"84t - 4.05 1.87 1.87

+ 261 d8 + 6:56 YCH) = 0 lor, 9"(t) + 2.614'(4) +6.1505 YCH) = Taking habelace transform on both side hp yoxt)} +2.61hf y'(t)} +6.625 hf y(D) = 0 oss?y(s) -sy(0) - y'0 +2.6180461) - Y603 +6.5FYCH) Or, spy(s)--1+2. GI(A463)-1) +6.55463) = 0 ot, f524BB9 +6-55344 : 460=1, H2O) = 1] Or, ($?+ 2.618 +6.55) 4(4) = +3.61 or, y(s) = A 2016+ 6.85 + win 3.61 SP+2.615 +6.55 SP+2.611 +6-55 or, 463) = za 403-1783. + ' te 50 86-85+1-703-1-703 3.61 s?+2.618 +6.55+1.703-1.703 08 YCA) = __ s t _3.61 (5+1-305) ?44.847. (8+1.305)?+ 4-847 = S+1:305 7.305 (0+1-305)2 +2.2012 (0+1-305)2+(2.2012 or, 4(s)

1.305.+ inverse daplaa transform on both side yld) = hy y()} = 1-4_$+1:30,5 + 2.30$_] (+13@5)2+(9.2012 (5+1.305) ?(2.2017 yld) = 23os tcos 2.204 + 2.305215084.im 2.200 2.20 yld) = "3054 [cos?.200 +2.305 din 2-2od] Note: 1.53 ha } - et 2.4'y *} = 1 3.49% the } =ēlocosbad 4. k'Prokto8} - teatambut,