Question

Solve the given integral equation or integro-differential equation for y(t). t y'(t)+ fit-vy(v) dv = 10t, y(0)=0 yola y(t) =

Answer 1

t * 0 Given Equation is; y'+)+ sit-vigevjdve lot lising the defini pion of Convolution, of Convolution, can be re-written as. 7' (t) + y*t Taking Laplace trans from on both sides wei have L {y'}+L {Y*t} lot or, = 10 £{t), SL143 - Hotels.Est]. = 10 Rit), I To or , S L. {Y}+ LRY] S2 S& or to (sº+1).£18). The 52 10 L {y} 10 S 341. (S+V(S2S+U). 10 .A S+1 + fet BS+G. 525+) 2 IS+(S&Stu YO Als&_S+1)+(3+1)(BS+G). Als?$++) + BS2++4J5 +G. - (A+BJS?+18+G-AJS +A+G) A+B=0 A+G = 10 B+G-A = 0 0 + =) 2 A+B+G= 10 =) B+.G = 10-2Ą From ② and ③ we get 10-2A-A=0 E) A = 1/3. from 0 We have from We have G = 10-10- > B= -1%3 20 3"

10 (S+V(SS+1) 10 3 St] + - S + St. s2_S+J ** * ㅗ나 - St 2. 10 3 + 10 3 Sti S-4 from ** and *** We have. -S+2 =[RST L.3] = + LG-A- 4 e. S+ E ISA ~ yt) 3 -S+%, 10 vo - t +1, e e L 국 s2+4. 3 } S +3 t e t 2 12 st 2. +g.e 3 3 Yzt {} is () + 을 를 5m() 2 is logy- bo ( 10 -t 3 4 2 + t 빠 10 e 3