Response to Exponential Inputs

Exponential

Author

Jasper Day

1 Response to Exponential Input

We start with an equation of form

\frac{dy}{dt} = ay + e^{st} \tag{1}

The output, y(t), is called the “exponential response.”

We have a starting deposit (initial condition) y = y(0) at time t = 0, and additional deposits equalling e^{st}.

With Equation 1, the solutions will be a multiple of e^{st}, written

y_p = Ye^{st} \tag{2}

(y_p is a particular solution of y.)

So we can substitute Equation 2 into Equation 1 and solve for Y.

\begin{gathered} Yse^{st} = aYe^{st} + e^{st} \\ (s-a)Y = 1 Y = \frac{1}{s-a} \end{gathered}

We’ve found a particular solution, but to find the complete set of solutions, we need to add all null solutions, where the input is zero.

\begin{gathered} \frac{dy}{dt} = ay \\ y = e^at \end{gathered}

So the complete set of solutions y_p + y_n is:

y(t) = \frac{e^{st}}{s-a} + Ce^{at}

and

y(0) = \frac{1}{s-a} + C

The complete solution that satisfies the initial conditions is:

\begin{aligned} y(t) &= \frac{e^{st}}{s-a} + \left[ y(0) - \frac{1}{s-a} \right]e^{at} \\ &= \frac{e^{st}-e^{at}}{s-a} + y(0)e^{at} \end{aligned}

y(0)e^{at} is the term growing out of the initial deposit (to use the money-in-the-bank scenario again), and \frac{e^{st}-e^{at}}{s-a} is the term growing out of the additional deposits.

\frac{e^{st}-e^{at}}{s-a} is the “very particular solution”, since it equals 0 at t=0, giving us the form y_{vp} + y_n.

1.1 Resonance

There is one problem: what if s=a?

Then you have resonance: you are putting money in with the same exponential as the natural growth of the money, and our formula has to change. The very particular solution will then be \frac{0}{0}: the formula has broken down.

So IF s=a,

y_{vp} + y_n = te^{at} + y(0)e^{at} \tag{3}

How do we get the resonant answer? We use l’Hôpital’s rule.

\begin{aligned} \lim_{s\to a} \frac{e^{st}-e^{at}}{s-a} &= \frac{\frac{d}{ds}\left(e^{st} - e^{at} \right)}{\frac{d}{ds}(s-a)} \\ &= \frac{te^{st}}{1} \\ &= te^{at} \end{aligned} \tag{4}

1

  • 1 Clearly, Equation 4 works whether you take the derivative with respect to s or with respect to a.

    • 1.2 Sum

    So, to recap:

    1. Find a particular solution y_p
    2. Find the nullspace, all solutions where the input (in this case e^st equals zero)
    3. Combine the particular and null solutions, then find an equation in terms of y(0)
    4. Check for resonance