Question

this is a differential equation question

please solve them with steps and details also, make sure it is correct

1. (3 points each) Find the following functions, (a) C{(t - 1) step(t - 4)} (b) C-{ S-2 } 2s +1 (c) C-{ s2 + 6s + 25

Answer 1

Ans. a L{t-1) -1² step (t (t-w? Heue this can be written as Lfct-12 ult-4) Now Heue we know that the Heue Lfelt-a) uſt-as} ce se L{f(t)} - 0 L& gets ult-a)} esa L{ getta)} - g In this case the inputs are not match Proof: let hlt-a) g(t) Now the condition becomes Lf net-a) ult-a) L {n(t)} [ from [from equation =) e e- o 2

Now het-a) = g(+) hlt-eta) getta) h(t) = g(t+a) Now put the value in equation ® e-as e L{getras Now come to the main Question which says L (t-1)2 u( t-w) inputs Heue the are not same so let h n(t-1) = g(+) h (t-1+5) = g(++5) h(t-4) Now the inputs are same so we get - L (++ se u(t-u)? g (t+5) = ) L { uſt-4)?

-) eto Now al peu equation 20 we get (tts) u(t-l) =) els 2 (45 t 4eet+2) { teſ + L{o} + L{2s e-45 키 2, +oL{t} + + 2S {1} S3 eus [5 + lo () + 25 2s ( ) es 2. + + s3 25 S 02 Ans.

6 tte} S-2 Heue we know that the t'{eet} - Htt-a) f(t-e) where H(t) is a Heaviside step function and heue a=4 - {e" (} .td-) {t} - How find i { -2 ? We know that 네 인 S-9 { } e eet 5 S-2 Now put the valued in equation 0 so we get

H(t-u) L 2 [ feruo (5-2)} L'de hay 2(t-4) H(4-4) e Ans. R I 25+1 82 + 65 +25 Now this can be written as 2 s tl 2 S + 6-5 = S2 +65 + 25 52 +65 + 9 + 16 = (25+ +6) -5 (S+ 3)2 +16 (25+6) (5+3)2 +16 S (5+3)2 +16 - 2 S. (s +3) (5+3)2 +16 1 (5+3)2 +16

Now اتا {ation Now By the lineauity propelity of inveuse laplace transform given t{a.fun a. ft) + b. gro } - a " { fro} + . C°fg of 의 so - { 2. St3 (+3)- +11 - S, (G+3+1G Now solue 네 2. S+ (J+3) +14 =) 2너 +3 2 다. (543)2 +16 Now we know that 네 gta -9t 인 cos(bt) (19+9)2462

so =) 2 니 S +3 (5+3)2 + (4) = ) 2. Terst cos(ht z) 2 e 3t cos (4t) Now solve S. (5 +372 +16 ایده ) 5 to Constantin (5+3)2 +16 Now we we know that Le sin wt} es s2+w2 Now Here we also know that L"! { f (sta)} = eat pct) ["{cotobetice} { otra} - t e

Now solve ["Antro] t"{ outle] = ) Heue L{ einutta se wa (5>0) = ) Helle w2=16 - 4 Now w S2+02 (1) s² + +42 [ حم 4 "La port"(time $ 159 { the nes 2 I sin (4t) 4 Now put the values in equation 0 so we get t"{ content e-3t Х (5+3)2 + 16 * sin (46) 11 -3t ୧ sin (46)

Now put all the values in equation 2 Jo We get 네 2 S+3 (S+3) + G (+3) + Ic }-{ srfant U 2. 드 S+3 (G+3) + 1G = 2e -3t os(4t) 3t sin(4t) Ans.