Question

6) 17pt) Solve the following IVP. y" +4y = f(t) y(O) = 0 ;'(0) = 0 if t <3 = (exp(0.2t) if t23

wondering how to solve this using the Laplace transformation. thank you

Answer 1

If you have any doutbs in the solution please ask me in commments ...here i used laplace method to solve the ivp and crossed check it also

** To solve the following IVP, y" + 4y = f1t) y(0) = 0, y'(0) = 0 if t <3 fit) es ; exp 10.2 t) ; if to3 O NOW as we know that ; if t zo ; if t>c ult-C) = 1 so f(t) = exp(0.2t) Ult-3) t/5 ult-3) f(t) e

so the given IVP will be, 775 y'' + uy e Ult-3) 4(0)-0, y'0) = 0 Lly) = y(s). NOW By apply Laplace Both side assume L(4") + ULIY) = Lle tis ult-3)). S2 L(y) – sylo) - y0) + Ully) = e-35 let) .: Lly'') = s? Lly) - Sy10) - YO) :11 Uelt) g(t)) = e-cs 219C++C)) y'(o)=0, 7(0) = 0 S2 yıs) + 4715) 35 -35 e-3 t+3 315 ile ) е d (e45) e dleat) - then By S-a -35 315 1 92 y(s) + 4 y(s) = e e S GE -35 > Vis) = e 315 5-) (sº+4) Let we assume (y(s)) inverse dapiace tremspor motion of y(s) = y(t) 910) which is solution of ow IVP e-35. 0315 (5-3) (52+4) 2-(419)) 2" Now L! e-35 y(t) (5-5) (52+4) us ) as we know that elle-ct F(S)) : U(t-c) flt-c) where flt) = ('(F(S)). we have By after using 315 ult-3) f17-3). ylt) = metoden where flt)= { e (64+4) 5:)

flt) = la' (152+4) (5-5) А + (52+4)(5-5) Bst C (S- S?+ 4 As?+ 4A + Bs? + CS - see A+B=0, § -01, 6-= 1 ЧА - = By compare → B = 20 A-C = 5 100A-50 = 25 50 A +50=0 A +5C = 0 -25 A = 25 101 B = ) -5 101 101 25 101 25 5+5 (101) (s2+4) (52+4)(5-5) f(t) = (s) pa" ( (5)) - ( 254 4,5) 4" (5-3) 5 25 1-' (*2) 1 101 {"(12) flt) 25 101 202 is 5 → FH) = 25 101 e sinet 25 eos 2t 101 202 50 et's 50 cos 2t s sin at = f(t) 202 -as 2-'(sta) l'(a) = cosat ; B (540) - Sinat t/s _315 e 50 cos 2(t-3) - 5 sin2(t-3) fit-3) = 50 e 10) 202 SO By -315 315 . {so e etis e - 50 cos 21t-3) -5 Sin 2 (7-3)} ylt) - u(t-3) e 202

so the solution of the given IVP; +15 25 e e+315 es 25 101 cos 2 (4-3) + 5 sin sin 2(t-3) ylt) = +-3). 202 101 Answer