Question

7- Solve the following differential equation foperiytz_ 64 - a) uo – a) -e* 1{10} useful for this problem. y" + y = (1-21), y(0) - 0, y'(0) - 1

Answer 1

Pb) solve the differential equation Sul" : y"+y slt-201) - and here given that 410) = 0, y'10) - 1 taking caplace Transform of egn 0 in both sides 4 y") + Lly) = 4(5(+-25)) Now we know that Lly") = 825 1+) - Sylo) - y.(o)= s? 5 Lt) - | Lly) = ☺ (+) L($(t-c))=ē CS 44s(+-201)) = 2 ans Now we we put Kalues in above equation s29(+)-1 +517) -QIS e e a (52+1) s?+1) 9 (+) -2 TTS e +1 9 (+) = - 2 TS e + S2+1 S2+1 taking Inverse laplace transform of above equation (lt)) = I'/ 2205 + L' S? 52+1 we know that that '(94)=74 TTS ylt) = id + [ $2+1 5+1

we know that El I sin sinat 52792 so = sint s?+12 we know second shifting property of Inverse Laplace l' [e-as fls) ] = f(t-a) ult-a) where L(flt)) = F(S) So Now e [" = sin(t-217). s(t-211 ) S2+1 Now we put values from egn ③ & 7 in egn 2 y(t) = sin(t-2TT).s(t-2) + Sint y (t) = sin(t-2).$(+-271) + sint Answer