Laplace Transform: Find Ratio $G(s)$ and value of $y(t)$ as $t\to \infty$

Question

Given the following equation

$ \dddot y + 3\ddot y +5\dot y = \dddot x+4\ddot x +6\dot x +8x $

Find $G(s) = \frac{Y(s)}{X(s)}$, when all initial conditions are zero.

As such I have gotten the Left Hand Side to be;

$(s^3 + 3s^2 +5s) Y(s)$

and the RHS to be;

$(s^3 + 4s^2 +6s+8) X(s)$

I am trying to find $x(t)$ but am having trouble progressing to this point as I am unsure if my work is correct and then the next relevant steps.

The next part involves finding the value of $y(t)$ as t $\implies \infty$, when $x(t)=\delta(t)$

I would appreciate any help in understanding the workings of this question.

Answer 1

So you found \begin{align} G(s)&=\frac{s^3+4s^2+6s+8}{s^3+3s^2+5s} \\ &=1+\frac{s^2+s+8}{s^3+3s^2+5s}=1+\frac8{5s}-\frac{3s+19}{5(s^2+3s+5)} \\ &=1+\frac8{5s}-\frac{3(s+\frac32)+\frac{29}2}{5((s+\frac32)^2+\frac{11}4)} \\ \end{align} So in the second task you get a sum of a delta spike, a jump at zero to the constant $\frac85$, and an oscillation with frequency $\frac{\sqrt{11}}2$ that declines exponentially in amplitude as $e^{-\frac32t}$.