Question

Solve the following IVPs using Laplace Transform: 2) y" + 4y' + 3y = 3 ezt; y(0) = y'(0) = 0

Answer 1

S-2 3 S-2 S-2 3 S-2 y"+4y'+3y = 3e24 Applying laplace transform on both sides L(y"+4y'+3y) = 2(322) 3 L(y")+44(y')+32(y) = (sºL(y)-sy(0)- y(0)} +4{s[(y)-y(0)}+3L (Y) = 3 sºL(y)-sy(0)- y'(0) + 4s[(y) - 4y(0) +3L(y) = y(O) = 0, y'(0) = 0 3 sºL(y)-s(0) - 0+4sL(y)-4(0) +3L(y) -2 (52 +48+3)Z(y) - 0+0-1) = 3 (s+1)(s+3)L(y) = 3 L(y) = (5-2)(s+1)(s +3) 3 L(y) = 5(8-2) 2(s+1)* 10(8+3) Applying Inverse laplace Transform on both sides,we get 3 y(t) = {-} 5(s-2) 2(5+1) ' 10(s+3) 3 I' +1 10 3 2 10 S-2 1 1 1 2(6+15+ *10(673 1 »(t)= (2+1) + S-2 y(t) 1 21 - = +