- #1

s3a

- 818

- 8

- Homework Statement
- Write the differential equation that is mathematically equivalent to the transfer function below: G(s) = Y(s) / R(s).

G(s) = C(s) / R(s) = (s^4 + 2s^3 + 5s^2 + s + 1) / (s^5 + 3s^4 + 2s^3 + 4s^2 + 5s + 2)

- Relevant Equations
- • ##G(s) = Y(s) / R(s)##

• ##r(t) = t^3 u(t)##

I have the solution to the problem, and I mechanically, but not theoretically (basically, why do the C(s) and R(s) disappear?), understand how we go from

##(s^5 + 3s^4 + 2s^3 + 4s^2 + 5s + 2) C(s) = (s^4 + 2s^3 + 5s^2 + s + 1) R(s)##

to

##c^{(5)}(t) + 3c^{(4)}(t) + 2c^{(3)}(t) + 4c^{(2)}(t) + 5c^{(1)}(t) + 2c^{(0)}(t) = r^{(4)}(t) + 2r^{(3)}(t) + 5r^{(2)}(t) + r^{(1)}(t) + r^{(0)}(t)##.

And, then I understand that the ##r(t)## needs to be replaced by ##t^3 u(t)##, but the final answer in the solution is

##c^{(5)}(t) + 3c^{(4)}(t) + 2c^{(3)}(t) + 4c^{(2)}(t) + 5c^{(1)}(t) + 2c^{(0)}(t) = 18δ(t) + (36 + 90t + 9t^2 + 3t^3) u(t)##, and I don't understand the step that leads the previous step to this final answer.

Could someone please help me figure out what it is that the solution did?

Any input would be GREATLY appreciated!

##(s^5 + 3s^4 + 2s^3 + 4s^2 + 5s + 2) C(s) = (s^4 + 2s^3 + 5s^2 + s + 1) R(s)##

to

##c^{(5)}(t) + 3c^{(4)}(t) + 2c^{(3)}(t) + 4c^{(2)}(t) + 5c^{(1)}(t) + 2c^{(0)}(t) = r^{(4)}(t) + 2r^{(3)}(t) + 5r^{(2)}(t) + r^{(1)}(t) + r^{(0)}(t)##.

And, then I understand that the ##r(t)## needs to be replaced by ##t^3 u(t)##, but the final answer in the solution is

##c^{(5)}(t) + 3c^{(4)}(t) + 2c^{(3)}(t) + 4c^{(2)}(t) + 5c^{(1)}(t) + 2c^{(0)}(t) = 18δ(t) + (36 + 90t + 9t^2 + 3t^3) u(t)##, and I don't understand the step that leads the previous step to this final answer.

Could someone please help me figure out what it is that the solution did?

Any input would be GREATLY appreciated!