Question

the and Convolution to find Use Laplace transformt solution to the + % IVP y" 24'+ 5y = s y lozo, y/o = -2

Answer 1

2/3 t - Given that y" - 2y + 5y yo)=0 y'(o)= -2 uring loplace tonoforon. Lff (t) = f(s) = f(t) e set det from formula Lyl - SLLY] - YO) g" = (9) L[y"] = SL[y'] - Y'CO) s2 L[y] - SYCO) - Y'(o) since from the eqreation L[y'] () اور تم (4) 11 [ Ye St]. - У ) е S+ (Sud SX [0-9] + SLłyJ-go) Yd) est at slonilarly L[y" S?[Y] - SYO - YO

on both side in Now applying Laplace transform eqreation a L[Eg" - 29' + sy] = L(42] 8 L[y] - Syo) - gro) SLCYJ + YO) +5l[y] L[VJ= Y(S) SS13 82 y(s)-S Y(S) +59(5) -sy(s) +59 (S) -0+2+0= 5 (8-5+5) ys) = 52 - 2 S y®) = (3/3 35/ (5²-5+5) (52-5+5) Now taking interne Laplace transform [![43)] - Y) - % 55/3 (52-5+5) 22515 om 1 2141

-> 52-5+5) opplying convolution theorem → 1?$£$) esy=[-$4893x C- Sars)3 Now applying Here leat f(3) = 1513 and rs)= 5513 S²S+5 563 5573 13 [ Sences} s²-SAS W 2 x S x 1 + 1 - 1 +5) = [t (mg [8-4-1 2 et sin ( a ) Jig

15/3 2/3 26 1 24 ein (5+) $513 ( 82-5+5) والے Now YA) = 24 24 3 Y sin nig sin ( t ) - 2x2 e ainting at Vagin ze pin (E ) () ak

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