Math Insight

Single autonomous differential equation problems

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  1. Consider the differential equation \begin{align*} \diff{ z }{ t } &= -8 z. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $z(0) = -6.3$?

  2. 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?

  3. Consider the dynamical system \begin{align*} \diff{ z }{t} &= -3 \left(z + 7\right) \left(z + 5\right) \left(z - 2\right)\\ z(0) & = z_0, \end{align*} where $z_0$ is an initial condition. Find all equilibria and analytically determine their stability. Use this information to draw a phase line diagram, including equilibria (using closed circles for stable equilibria and open circles for unstable equilibria) and vector field. Then, sketch a graph of the solutions corresponding to the equilibria (using solid lines for stable equilibria and dashed lines for unstable equilibria) and solutions for initial conditions $z_0 = -9.1, -6.0, -3.6, 5.2$.

  4. For the dynamical system $ z'(t) = f(z, c)$, where the function $f$ of $z$ also depends on a parameter $c$, a bifurcation diagram with respect to the parameter $c$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $c$, determine the number of equilibria, their values, rounded to the nearest integer, and their stability. For each case, sketch the phase line, including equilibria and vector field. Use a solid circle for stable equilibria and an open circle for unstable equilibria.
      1. $c= -8$
      2. $c= -1$
      3. $c= 3$
    2. Identify any bifurcation points.

  5. For the dynamical system $ \diff{ x }{t} = f(x,\beta),$ the function $f$ of $x$ depends on a parameter $\beta$, as shown in the graphs for $\beta=-6, -3, -2, -1$, below. For values of $\beta$ in between those shown, $f$ changes smoothly, so its graph will be somewhere in between the snapshots shown. Sketch a bifurcation diagram with respect to the parameter $\beta$, for $-6 \le \beta \le -1$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $\beta=-6$

    $\beta=-2$

    $\beta=-3$

    $\beta=-1$

  6. Consider the dynamical system \begin{align*} \diff{y}{t} = 3(y-2)(y-1)(y+1). \end{align*}
    1. Find the equilibria and use the stability theorem to calculate their stability.
    2. Sketch the vector field on the phase line illustrating the rate of change $\diff{y}{t}$.
    3. Graph the solution $y(t)$
      1. for the initial conditions $y(0)=1.5$.
      2. for the initial conditions $y(0)=2$.
      3. for the initial conditions $y(0)=2.5$.

  7. Estimate the solution of the differential equation \begin{align*} \diff{ x }{t} &= - 2 \ln{\left (1.4 x^{2} \right )}\\ x(0) &= -0.1, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.5$ to estimate $x(1.5)$.

  8. For the differential equation \begin{align*} \diff{q}{t} = e^{-q^2-3q+1} \end{align*} find the equilibria and use the stability theorem to calculate their stability. Graph the solution for the initial condition $q(0)=0$.


  9. Consider the dynamical system \begin{align*} \diff{ x }{t} &= f(x) \end{align*} where the function $f$ is graphed to the right.
    1. Find all equilibria.
    2. For each equilibrium, determine its stability by sketching the solution for initial conditions just above and below the equilibrium.