plz help ty s solve the ven n t a value problem Solutiony 2

plz help ty

Solution

y\'\' + 2y\' + 2y= 0 ; y(0) =1 and y\'(0) = 0

taking laplace transform of above Differntial Equation

S²Y(S) - SY(o)-Y\'(0) + 2 * (SY(S) - y(0) ) +2Y(S) = 0

S²Y(S) - S + 2 * (SY(S) - 1 ) +2Y(S) = 0

(S² + 2S +2 ) Y(S) = S + 2

Y(S) = ( S + 2 ) / ( S² + 2S +2 )

Y(S) = (S+2) / ( (S+1)² + 1 )

Taking inverse laplace transform

y(t) = e^-t L^-1 { (S+1) / ( S² + 1 )

= e^-t L^-1 { S/ ( S² + 1 ) + 1 / ( S² + 1   ) }

= e^-t ( cos t + sin t )