Question

Solve y''+9y=f(t) with y(0)=0 and y'(0)=0 and f(t)=sin(t) when $t \in [0, \pi]$ and zero otherwise

Use laplace transform

Answer 1

o tot Y" + Gy = f(t) 17)-S Sint 05t< 910-0, 4'60)=0 f(t) = Sint- Lly" + gy) = L( f(t)) 219" +9 hly) = 4( Sint - UCE-T) Sint) (s*y(s)-sy(0) -y')) + gy(s) = b[Sint) blult-71) Sint) SMS) - e-TS LSin (t+7)) Y(s) (52+9)= - -ts (-sin(t)) S²+1 | Yls) (52+9)= + e-TTS ( sin (t)) Sa+1 Y(s) (52+9) - + e-tis S²+1 S²+1 Y15) - te-Tis (52+1 (54+9) (52+1)(54+9) Ys) - ta - toe te-TS - €"S - 8 (5²+1) Scanned with CamScanner 8(5²9) 865²+1) 815²+9) !

L"(Y(s)) = 41965.) - _ yle) sin lt) - 1 Sin(38) tft sin(-7) sen (314-))) UCE-1) 24 I dit) - Sinit) Ult- Sinlt) - Sin (3+) 8 २५ +! Sin (3+) Ult-T) 24 Using hlsin (at2) = a s²ta2 hlult-ef1t-c)) = a(flt)) CS$canned with CamScanner