Math Insight

Single autonomous differential equation problems

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  1. For the differential equation \begin{align*} \diff{z}{t} = h(z), \end{align*} the function $h(z)$ is graphed below. Autonomous differential equation example function 2
    1. Sketch the vector field illustrating the rate of change $\diff{z}{t}$.
    2. Find the equilibria and calculate their stability.
    3. Graph the solution $z(t)$
      1. for initial condition $z(0)=3.8$.
      2. for initial condition $z(0)=4.2$.

  2. Estimate the solution of the differential equation \begin{align*} \diff{ x }{t} &= 6 \ln{\left (0.8 x^{2} \right )}\\ x(0) &= -0.4, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.2$ to estimate $x(0.6)$.

  3. For the dynamical system \begin{align*} \diff{ u }{t} = g(u, a), \end{align*} where the function $g$ of $u$ also depends on a parameter $a$, a bifurcation diagram with respect to the parameter $a$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $a$, determine the number of equilibria, their values, rounded to the nearest integer, and their stability. Sketch the phase line, including equilibria. Use a solid circle for stable equilibria and an open circle for unstable equilibria.
      1. $a= -5$
      2. $a= 2$
      3. $a= 10$

    2. Identify any bifurcation points.

  4. For the dynamical system $ \diff{ v }{t} = g(v,\gamma),$ the function $g$ of $v$ depends on a parameter $\gamma$, as shown in the graphs for $\gamma=0, 5, 10, 25$, below. For values of $\gamma$ in between those shown, $g$ changes smoothly, so its graph will be somewhere in between the snapshots shown. Sketch a bifurcation diagram with respect to the parameter $\gamma$, for $0 \le \gamma \le 25$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $\gamma=0$

    $\gamma=10$

    $\gamma=5$

    $\gamma=25$

  5. Consider the differential equation \begin{align*} \diff{ x }{ t } &= 9.6 x. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $x(0) = -4.8$?

  6. For the dynamical system \begin{align*} \diff{r}{t} = -re^{r} \end{align*} find the equilibria and use the stability theorem to calculate their stability. Graph the solution for the initial condition $r(0)=5$.

  7. For the dynamical system \begin{align*} \diff{ z }{t} &= \left(2 z^{2} + 2 z\right) e^{- z}, \end{align*} find all equilibria and analytically determine their stability (i.e., use the stability theorem). Using this information, draw a phase line diagram with equilibria and vector field. (Be sure to indicate the stability of the equilibria.)

  8. Consider the dynamical system \begin{align*} \diff{u}{t} = u(2-u). \end{align*}
    1. Using any valid method, determine the equilibria of the dynamical system and their stability.
    2. Graph of the solution of the dynamical system with initial condition $u(0)=0.8$.
    3. Use the Forward Euler algorithm with time step $\Delta t=2$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(2)$, $u(4)$, $u(6)$, and $u(8)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?
    4. Use the Forward Euler algorithm with time step $\Delta t=1$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(1)$, $u(2)$, $u(3)$ and $u(4)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?
    5. Use the Forward Euler algorithm with time step $\Delta t=0.5$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(1/2)$, $u(1)$, $u(3/2)$ and $u(2)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?

  9. Consider the differential equation \begin{align*} \diff{z}{t} = -3(z-5)(z-10). \end{align*}
    1. Find the equilibria and use the stability theorem to calculate their stability.
    2. Sketch the vector field illustrating the rate of change $\diff{z}{t}$.
    3. Graph the solution $z(t)$
      1. for the initial conditions $z(0)=0$.
      2. for the initial conditions $z(0)=8$.
      3. for the initial conditions $z(0)=12$.

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Math 201, Spring 22