Math Insight

Single autonomous differential equation problems

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  1. 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?

  2. Consider the dynamical system \begin{align*} \diff{ s }{t} &= -3 \left(s + 8\right) \left(s + 1\right) \left(s - 3\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 $s_0 = -9.2, -3.1, 0.6, 5.1$.

  3. 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 $z(0)=12$.

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

  6. For the dynamical system \begin{align*} \diff{ y }{t} = f(y, \alpha), \end{align*} where the function $f$ of $y$ also depends on a parameter $\alpha$, a bifurcation diagram with respect to the parameter $\alpha$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $\alpha$, 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. $\alpha= -10$
      2. $\alpha= 1$
      3. $\alpha= 10$

    2. Identify any bifurcation points.

  7. For the dynamical system $ \diff{ s }{t} = g(s,c),$ the function $g$ of $s$ depends on a parameter $c$, as shown in the graphs for $c=-1, 2, 3, 4$, below. For values of $c$ 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 $c$, for $-1 \le c \le 4$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $c=-1$

    $c=3$

    $c=2$

    $c=4$

  8. Consider the differential equation \begin{align*} \diff{ w }{ t } &= -5.5 w. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $w(0) = -4.9$?

  9. Estimate the solution of the differential equation \begin{align*} \diff{ u }{t} &= 3 e^{- 1.0 u} + 6\\ u(0) &= 0.8, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.3$ to estimate $u(1.2)$.

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Math 1241, Fall 2020