DIFFERENTIAL EQUATIONS
SM212
Prof W D Joyner

  1. The SAGE version of the heating/cooling example CP assignment below can be turned in for extra credit.
  2. The SAGE version of the pendulum assignment below can be turned in for extra credit.
If you are interested in any of the others, please ask me.

  • Possible Maple/SAGE extra credit work. Included is an example of how the work is to be done.

    Example CP: Heating and cooling of buildings. (See Heating and cooling of buildings for a solution.)

    The following is a somewhat more general heating and cooling DE that includes additional heating and air conditioning of a building.


    \begin{displaymath} {{dT}\over{dt}}= K[M(t) - T(t)] + H(t) + U(t). \end{displaymath} (1)

    1. Suppose that $M(t) = 75^{\circ} + 20\sin (\pi t/12)$, $K = 2$, $H(t) = 75^{\circ}$, $U(t) = 0$. By use of a computer software package, find and plot solutions, where the initial condition is

      (a)
      $T(0) = 60^{\circ},$
      (b)
      $T(0) = 70^{\circ},$
      (c)
      $T(0) = 80^{\circ}$ over a 24 hour period.

    2. What do you observe about the effect of the initial condition on the solution as time progresses?

    3. Solve (1) explicitly and determine what factor in the solution might account for this behavior?

    4. Let $T(0) = 60^{\circ}$ and let the other values stay fixed except let $K = 2, 4, 6.$ Graph the three solutions for these three $K$ values.
    Can you describe the effect of the time constant $K$ has on the solutions?
    Extra credit maple assignments

    CP 1. Pendulum behavior. For a simple pendulum without an external driving force, the angle $\theta (t)$ the pendulum makes with the vertical is described by the DE

    \begin{displaymath} \theta'' + {{g}\over{L}}\sin \theta = 0, \end{displaymath} (2)

    where $g = 32.2 {\rm ft/sec^2},$ and $L$ is the length of the pendulum in ft. This needs to be solved by numerical means to get a good approximation.

    1. Let $L = 8$, $\theta (0) = \pi/4$, $\theta '(0) = 0$. For $0 \leq t \leq 5\pi,$ plot a solution to this pendulum problem.
    2. If you have had any experience with a pendulum, does this solution match you experience?

    CP 2. Springs with large displacements. Hooke's Law is an approximation for spring behavior that is only valid for small displacements. A more general DE for larger displacements has an alteration in the Hooke's Law term. In place of $kx,$ one writes $kx +\epsilon x^3,$ where $\epsilon>0$ for a ``hard spring'' and $\epsilon<0$ for a ``soft spring.'' For the non-forced case, this yields Duffing's equation


    \begin{displaymath} mx'' + \beta x' + kx +\epsilon x^3 = 0. \end{displaymath} (3)

    1. Use $m = k = 1, x(0)=x'(0) = 0$ and plot three different solutions for $\beta = 1, 2, 3.$ For each of these damping constants, make a graph of the solutions where $\epsilon = -0.5, 0, 0.5$ so that you can make a comparison between the graphs for each $\beta$ and each $\epsilon.$

    2. Try and explain from a physical viewpoint the differences in the graphs relative to the $\beta$ and the three values of $\epsilon$. Suppose that we add an external driving force $f(t) = F_0\cos (\gamma t)$ to equation (3). This gives


      \begin{displaymath} mx'' + \beta x' + kx +\epsilon x^3 = F_0\cos (\gamma t). \end{displaymath} (4)

    3. For $m = k = F_0 = 1$, $\epsilon = 5$, $\beta = 0.25$, $x(0) = 0$, $\gamma =1$ and window $0\leq t\leq 50$ graph in the same window solutions to (4) where $x'(0) = -5, 0, 5.$ Now repeat this for the cases, $\gamma = 2.5, 4.$

    4. Does it appear that the solutions are approaching a stead state?

    5. If so, estimate the amplitude of the steady state solution.

    6. Due to the $x^3$ term, this is not a linear differential equation. For linear DE's, the solutions and hence behavior being model tends to change gradually when the physical parameters and initial conditions are changed gradually. For non-linear equations, this need not be the case. When the behavior of a solution changes abruptly as a value of a parameter passes through a specific critical value, we get what is called a bifurcation. One of the most significant such bifurcations occurs when only the initial conditions are altered. To see an indication that the change from one steady state to another is a bifurcation under initial condition changes, plot two solutions of (4) with $m = k = F_0 = 1$, $\beta = 0.25$, $\epsilon = 5$, $\gamma = 2.5,$ and (a) $x(0) = 0.86, x'(0)=0,$ and (b) $x(0) = 0.87, x'(0) =0.$ Try a $20\leq t \leq 30$ domain.

    CP 3: An electrical circuit with a periodic emf

    Your assignment is to use SAGE to solve the DE Li'+Ri=E, where i=i(t) is the current and E=E(t) is the sawtooth function saw(t)=t-[t] (here [x] denotes the integer part of x, so [1.1]=1, [7/2]=3, for example). See sawtooth or
    sawtooth function for a picture.

    Graph your solution from t=0 to t=10 for L=1 henry and R=1,2 and 4 ohms.


    Prof David Joyner created 2003-09-14. Last modified 8-18-2008.