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Find the initial condition $\:y(0)\:$ such that the response of the system described by

$$ \dfrac{dy(t)}{dt}+3y(t)=x(t) $$ to the unit step input i.e., $\Bigl(x(t)=u(t)\Bigr)$ exhibits no transient behaviour.

I am trying to solve the above question and I am not sure how to prove if something has transient behaviour or not. Here is what I've done so far

$$(D^2 + D)y(t) = x(t)$$

so $D = 0$ or $-1$

Therefore, $y(t) = ce^{-t}$

However, since we aren't given initial conditions, I am not sure what to equate $y$ to.

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  • $\begingroup$ I guess if you give it the initial condition matching the asymptotic steady state solution then there will be no transients. $\endgroup$ – Maxim Umansky May 6 '16 at 0:45
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    $\begingroup$ This is pure mathematics. No physics at all. Have you tried Mathematics SE? $\endgroup$ – sammy gerbil May 6 '16 at 3:16
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I might be wrong, but shouldn't the equation be:

$(D + 3) y(t) = x(t)\Rightarrow$ auxiliary equation to be: $D+3 = x(t)$

Where $D = \frac {d}{dt}$ ?

Again, I am not sure.

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The homogeneous solution is actually $Ce^{-3t}$, I'm not sure where you got a $D^2$ from in a first order problem. Anyway, a particular solution is $u(t)/3$, because its derivative will vanish and so you need $3y=u$. So the general solution is $y=Ce^{-3t}+u(t)/3$ where $C$ depends on the initial condition. The asymptotic behavior is in $u(t)/3$ while the transient behavior is in $Ce^{-3t}$. Find $y(0)$ such that $C=0$.

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