Question

Solve the given integral equation or integro-differential equation for y(t). t y' (t) + y(t) - y(v) sin (t - v) dv = - 10 sint, y(0) = 10 Syru 0 y(t) =

Answer 1

solution ; 270) = 10 y' (+ 9 + 964) - Jasin (+-10) 74W) du = -10 Sint now using convolution theorem rheabone equn cambe writen as y '(t) + G) – F(4) * 2 t) - 10 sint F (+= sint (where +-) (& Ft) 3 = 1 F (+) do 3 S since-w3 yule taking Laplace with siek meget La' (t)} + 2430} – 28 ) BG)507 ཅན Now =10L a $1$@} - 700) + L5793} - 2.5 sine} L434) -10 L L{sine} SLER)} & 10th i 10 tuhhar 14gt 2{ (+)} 24o. I 1+32 키 Ls 0} (3+1 1792 750) -10 1492 (since 260)=10 )

-10 +10+105 (+5) (+35) less los 7 last losh 105 » LS 6)} ( (+14) at set 33 +145-1 3703} ( *** 435) its? 2236)} 33+5+5) esy(t)} 33+575 7 436)} 7: $ 567 {5 + 25 : +62 +1Gt 2230} (3 + 2) + 124 ? - (s + 2) - 1 (3 + 2) + 3 105 375 +1 s - 10 10 2 -10x2. 72 364} = 10 (5+ 3) (3+1) 7 (12) 2384419 670 7 L 70)} = 10 (sif) 7 (3) 536 Now taking eaplane inverse both side 105 3 + 2 210 L 17 L { 34} - (95) *

it (t) = 10 e 2 cos dat. cos sin dat since (3=100) et zinket 121 =e (8+137 ly(t) = 10 e cost lo e costat sin egt