Estimate time to reach steady state

[mpm]

I have some first order differential equations. I have found the steady state response for each equation. However, now I need to estimate how long it will take to reach steady state.

Can anyone tell me the formula for that or get me started on how i go about doing this. I'm not finding anywhere in my textbook where it says or shows how to do this.

 

[CarlB]

Try doing the following substitution on your DE. Say the function is y(x,t) and your steady state solution is s(x). Take your DE and replace y(x,t) with s(x)+u(x,t). Now your u(x,t) is the difference from the steady state.

To get an idea of how it changes with time, you might try, depending on your DE and whether this make sense, linearizing by eliminating all but the terms linear in u. That will probably allow you to put the DE into a form where an exponentially decaying sine wave is the solution. The answer could then be the rate of exponential decay in the sine wave.

That help?

Carl