#### PhillDubya

##### New Member

Equation:

d³/dt³ (y) + d²/dt² (y) + dy/dt + y = d²/dt² (u) + du/dt + u

Initial Conditions all = 0, and u(t) = unit step, and

I used the Laplace for solving it and got:

Y(s) = [(s^2 + s + 1) / (s^3 + s^2 + s +1)] U(s)

U(s) = unit step = 1/s

Y(s) = (s^2 + s + 1) / s(s^3 + s^2 + s +1) = s^2 + s +1/ s(s²+1)(s+1)

I am having problems converting this back (the inverse Laplace) in order to get a y(t) function , or back to the time domain.

1/s = 1, 1/s²+1 =sint, 1/s+1 = e^at However, I am assuming I cannot convert like this because of the numerator [(s^2 + s + 1)] which will not factor.

So, is partial fraction expansion the only option with this?

Would it be easier to just solver the 3rd order differential?

Thanks for your time.

d³/dt³ (y) + d²/dt² (y) + dy/dt + y = d²/dt² (u) + du/dt + u

Initial Conditions all = 0, and u(t) = unit step, and

**solve for y(t)**

I used the Laplace for solving it and got:

Y(s) = [(s^2 + s + 1) / (s^3 + s^2 + s +1)] U(s)

U(s) = unit step = 1/s

Y(s) = (s^2 + s + 1) / s(s^3 + s^2 + s +1) = s^2 + s +1/ s(s²+1)(s+1)

I am having problems converting this back (the inverse Laplace) in order to get a y(t) function , or back to the time domain.

1/s = 1, 1/s²+1 =sint, 1/s+1 = e^at However, I am assuming I cannot convert like this because of the numerator [(s^2 + s + 1)] which will not factor.

So, is partial fraction expansion the only option with this?

Would it be easier to just solver the 3rd order differential?

Thanks for your time.

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