Question

Use the Laplace transform to solve the given integral equation. f(t) = tet + S'ence- tf(t - t) dr f(t) =

Answer 1

Solution t fct) = tet+d T Ef (t-1)] di Apply laplace transform on both side of above integral equation = LCfct)) =4tet ] + L List f(t-1) dr Now, L ( tet) = (432 h Llet) -(-1) d si (۱) (5-102 o Lct et, (8-172 f By convolution theorem 14 16 + 964);= 1 gCt). f(4-1)dt) Lcfct)) · Lcgct) t Solfistfct-I) do] LCfa) · LCt) t Note . Here get=t. [/T fct-T) dT L (ft)). 32

% from ① , ② & G ILCAct)) = 1 LCfct)) (SH)2 2 =) L(Act)) - LCfct)) - c5-172 2 =) 1(f(+)) [1 - 5 (52 2 =) LCfc+)). (8-12 s² - 1 52 (8-12 (st) (S-1) Y8 + + LCf(t)) = cst (SH) Sou By partial fraction (3/4) + Y2 CH² (SH) (+) (842 (543 =) LC f(t)) = 4 (8-)?? -> let Lefrt)) = f(s) (18) SH) 8 s-l 2 cst 8 st) = f(s) = 16+ 2 (51) + 2 (5-13 8 cst)

Now sides Apply Laplace inverse on both of we get, U( Feli) = { ifal | Los cs13 8 co by Elst & [st fct) tette et il Het i te first shifting property = f(t) et ustett telee - frt) = Į [et tot et + 4t et - et] It is required solution of given Integral equation 8