Find the Lapplace transform of ytU2t cos 3t yty3tt1 Find the

Find the Lapplace transform of y(t)=U_2(t) cos 3t y(t)=y_3(t)(t-1) Find the inverce Laplace transform of Y(s)= s/s^2+6s+12

L(cos3t)= s/s^2+9

So using the formula

L(u_a(t) =e^(-as) L(f(t+a))

L(u_2(t) cos3t) = e^-2s L(cos(3t +3a) )

= e^-2s L(cos3t cos 3a - sin3t sin3a)

=e^-2s [cos3a s/(s^2+9 ) - sin3a 3/(s^2 +9)]

L(u_3(t)(t-1) ) = e^-3s(L(t+2)) = e^-3s (1/s^2 + 3/s)

s/s^2+6s+12 = s/((s+3)^2 +3) = s+3/((s+3)^2+ 3 ) - 3/((s+3)^2+ 3 )

inverse transform is therefore

e^(-3t) cos(sqrt(3) t) - e^(-3t) sqrt(3)sin(sqrt(3) t)