Math Insight

Single autonomous differential equation problems

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  1. For the dynamical system $ s'(t) = h(s, c)$, where the function $h$ of $s$ 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. $c= -9$
      2. $c= -1$
      3. $c= 3$
    2. Identify any bifurcation points.

  2. Consider the dynamical system \begin{align*} \diff{ x }{t} &= -2 \left(x + 7\right) \left(x + 4\right) \left(x + 0\right)\\ x(0) & = x_0, \end{align*} where $x_0$ is an initial condition. The graph of the function $f(x) = -2 \left(x + 7\right) \left(x + 4\right) \left(x + 0\right)$ is shown below. Use the graph to sketch the solution $x(t)$ for each of the following initial conditions: $x_0 = -7.6$, $x_0 = -7$, $x_0 = -5.2$, $x_0 = -4$, $x_0 = -1.2$, $x_0 = 0$, and $x_0=1.8$.

  3. 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?

  4. 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 $q(0)=0$.

  5. Consider the dynamical system \begin{align*} \diff{ s }{t} &= -1 \left(s + 4\right) \left(s + 0\right) \left(s - 8\right)\\ s(0) & = s_0, \end{align*} where $s_0$ 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 $s_0 = -9.4, -1.2, 3.2, 9.6$.

  6. Consider the differential equation \begin{align*} \diff{ x }{ t } &= 3.5 x. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $x(0) = 8.2$?

  7. Estimate the solution of the differential equation \begin{align*} \diff{ y }{t} &= 9 \ln{\left (y^{2} \right )}\\ y(0) &= -0.7, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.1$ to estimate $y(0.3)$.


  8. Consider the dynamical system \begin{align*} \diff{ y }{t} &= f(y)\\ y(0) & = y_0, \end{align*} where the function $f$ is graphed to the right and $y_0$ is an initial condition. For each of the following initial conditions, sketch the graph of the solution $y(t)$.
    1. $y_0 = -9.2$
    2. $y_0 = -4.0$
    3. $y_0 = 3.0$
    4. $y_0=9.6$

  9. For the dynamical system $ \diff{ y }{t} = f(y,b),$ the function $f$ of $y$ depends on a parameter $b$, as shown in the graphs for $b=-22, -7, -2, 3$, below. For values of $b$ 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 $b$, for $-22 \le b \le 3$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $b=-22$

    $b=-2$

    $b=-7$

    $b=3$

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Math 1241, Fall 2020