Math Insight

Single autonomous differential equation problems

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  1. Consider the differential equation \begin{align*} \diff{ v }{ t } &= 3.2 v. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $v(0) = -8.9$?

  2. Estimate the solution of the differential equation \begin{align*} \diff{ u }{t} &= - 2 \ln{\left (0.3 u^{2} \right )}\\ u(0) &= 0.2, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.5$ to estimate $u(1.5)$.

  3. 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 $q(0)=0$.

  4. 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 $x(0)=4.2$.

  5. For the dynamical system $ \diff{ s }{t} = f(s,\beta),$ the function $f$ of $s$ depends on a parameter $\beta$, as shown in the graphs for $\beta=-13, -10, -9, -8$, 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 $-13 \le \beta \le -8$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $\beta=-13$

    $\beta=-9$

    $\beta=-10$

    $\beta=-8$

  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. For the dynamical system \begin{align*} \diff{ x }{t} = g(x, c), \end{align*} where the function $g$ of $x$ 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. Sketch the phase line, including equilibria. Use a solid circle for stable equilibria and an open circle for unstable equilibria.
      1. $c= -9$
      2. $c= 4$
      3. $c= 9$

    2. Identify any bifurcation points.

  8. Consider the dynamical system \begin{align*} \diff{ y }{t} &= 3 \left(y + 2\right) \left(y + 0\right) \left(y - 7\right)\\ y(0) & = y_0, \end{align*} where $y_0$ 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 $y_0 = -9.2, -1.0, 3.5, 8.2$.

  9. 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?

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Math 1241, Fall 2020