Question

Find the complete time-domain solution y(t) for the rational algebraic output transform Y(s):_________

Answer 1

y(t)= 11/3 e^(-t) - 5/2 e^(-2t) -1/6 e^(-4t)

Explanation:

`Y(s)=(s+3)/((s^2+3s+2)(s+4)) + (s+3)/(s^2+3s+2) +(1)/(s^2+3s+2)`

We know that
`s^2+3s+2=(s+1)(s+2)`, so we have

`Y(s)=(s+3+(s+3)(s+4)+s+4)/((s+1)(s+2)(s+4))`

By using the method of partial fraction we have:

`Y(s)=(11)/(3(s+1)) - (5)/(2(s+2)) -(1)/(6(s+4))`

Now we have:

`y(t)=L^(-1)[Y(s)](t)`

Using linearity of inverse transform we get:

`y(t)=L^(-1)[(11)/(3(s+1))](t) -L^(-1)[(5)/(2(s+2))](t) -L^(-1)[(1)/(6(s+4))](t)`

Using the inverse transforms

`L^(-1)[c(1)/(s-a)]=ce^(at)`

we have:

`y(t)=11/3 e^(-t) - 5/2 e^(-2t) -1/6 e^(-4t)`