Question

Indicate which of the following options represents the solution of the equation with initial conditions y (0) = 0 and y ′ (0) = 5 with the equation:

g" +10g + 2lg = Us (t)

*GIVEN OPTIONS*:

Answer 1

y" + 10y' + 214 = Usct), 7(0) = 0,410): 5 Using laplace Tramsfroms Lhy" } + lochy'}+21 chy}= hh Us (t)} By propesty Lhyil= 5? Y(s) - Sy(0) -y (0) LÝ y'= Soy (5) – y(0) Lhy = Oy) and Lh Hult)}= S Now, y(0) = 0, y (O) = 5 Lhy"} = S²y(s) - 5 Lhy'} = SY(S) Lhy} = (s) Lhu 5 (t)} = e 5s 2 S 5 -55 => 524(5) - 5+105YCS) + 214() -5s Now, szy(5) PIOS Y (5) +21 4 (5) = 5 + ľ 5 (52+105 +21) Y(s) = 5+ (-55 S (s+3)(5+7) Y() = 5 + £ 55 S 5 + ess Y(s) (5+3)(5+2) 5(5+3)(s+7)

Now, resolving each into parshal frachins, we get Y(S) = 5 I* 4(+3) 4(5+1) 12(5+3) 5 + e-5 215 + 5 + = y(t) = 5 e -3t + 28(5+7) Now, y(t) = ['| 5 4463 14 (5+3) 4 (577) + essa 215 12 (5+3) 28 (+7) - 24 t + 21h e ss ( 21 12(5+3) 28(5+7) 26+7] Now, Recall t'{e-as F (5)} = f (t-a) Halt) whese flt) = ('{FC)} Now, F(S) = 215 12(5+3) 2815+7) f(+) = ze 3t+ e-7t f(t-5) = Le ,-3(t-5) -7(+-5) + 1 28 TI 21 12 28

قے 2. y Lt = 5 e 3t 5 et + us it) { 21 236-5) 4 , 2764-5}} sest set + 1/ 45t 12 ryct) 5 e 35-7 -- 1 2 3 4 5 (t) : e 15-3 15-3t + 1 Usit e 28 Frost option is correct