Math Insight

Single autonomous differential equation problems

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  1. Estimate the solution of the differential equation \begin{align*} \diff{ w }{t} &= - 3 e^{1.2 w} + 8\\ w(0) &= 0.7, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.2$ to estimate $w(0.8)$.

  2. For the dynamical system $ s'(t) = h(s, \gamma)$, where the function $h$ of $s$ also depends on a parameter $\gamma$, a bifurcation diagram with respect to the parameter $\gamma$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $\gamma$, 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. $\gamma= -10$
      2. $\gamma= -1$
      3. $\gamma= 7$
    2. Identify any bifurcation points.

  3. Consider the differential equation \begin{align*} \diff{ v }{ t } &= -7.9 v. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $v(0) = -2.9$?


  4. Consider the dynamical system \begin{align*} \diff{ s }{t} &= f(s)\\ s(0) & = s_0, \end{align*} where the function $f$ is graphed to the right and $s_0$ is an initial condition. For each of the following initial conditions, sketch the graph of the solution $s(t)$.
    1. $s_0 = -9.0$
    2. $s_0 = -4.2$
    3. $s_0 = 5.4$
    4. $s_0=9.7$

  5. 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 $q(0)=0$.

  6. 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?

  7. Consider the dynamical system \begin{align*} \diff{ x }{t} &= -4 \left(x + 4\right) \left(x - 4\right) \left(x - 6\right)\\ x(0) & = x_0, \end{align*} where $x_0$ 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 $x_0 = -8.8, -0.8, 5.4, 7.2$.

  8. For the dynamical system $ \diff{ v }{t} = f(v,\gamma),$ the function $f$ of $v$ depends on a parameter $\gamma$, as shown in the graphs for $\gamma=-20, -11, -8, -5$, below. For values of $\gamma$ 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 $\gamma$, for $-20 \le \gamma \le -5$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $\gamma=-20$

    $\gamma=-8$

    $\gamma=-11$

    $\gamma=-5$

  9. Consider the dynamical system \begin{align*} \diff{ s }{t} &= -3 \left(s + 0\right) \left(s - 5\right) \left(s - 8\right)\\ s(0) & = s_0, \end{align*} where $s_0$ is an initial condition. The graph of the function $f(s) = -3 \left(s + 0\right) \left(s - 5\right) \left(s - 8\right)$ is shown below. Use the graph to sketch the solution $s(t)$ for each of the following initial conditions: $s_0 = -0.8$, $s_0 = 0$, $s_0 = 3.5$, $s_0 = 5$, $s_0 = 6.5$, $s_0 = 8$, and $s_0=9.6$.

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Math 1241, Fall 2020