Math Insight

Single autonomous differential equation problems

Name:
Group members:
Section:

  1. Consider the dynamical system \begin{align*} \diff{ u }{t} &= f(u)\\ u(0) & = u_0, \end{align*} where the function $f$ is graphed to the right and $u_0$ is an initial condition. For each of the following initial conditions, sketch the graph of the solution $u(t)$.
    1. $u_0 = -9.6$
    2. $u_0 = -5.0$
    3. $u_0 = -0.4$
    4. $u_0=5.2$

  2. Consider the dynamical system \begin{align*} \diff{ y }{t} &= 2 \left(y + 8\right) \left(y + 3\right) \left(y + 1\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 $y_0 = -9.6, -6.5, -1.4, 4.5$.

  3. Consider the differential equation \begin{align*} \diff{ x }{ t } &= -6.7 x. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $x(0) = 2.2$?

  4. For the dynamical system \begin{align*} \diff{r}{t} = -re^{r} \end{align*} find the equilibria and use the stability theorem to calculate their stability. Graph the solution for the initial condition $r(0)=5$.

  5. 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 $y(0)=2.5$.

  6. For the dynamical system $ s'(t) = h(s, b)$, where the function $h$ of $s$ also depends on a parameter $b$, a bifurcation diagram with respect to the parameter $b$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $b$, 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. $b= -10$
      2. $b= -1$
      3. $b= 9$
    2. Identify any bifurcation points.

  7. Estimate the solution of the differential equation \begin{align*} \diff{ y }{t} &= 8 \ln{\left (0.6 y^{2} \right )}\\ y(0) &= -1, \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{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?

  9. For the dynamical system $ \diff{ y }{t} = g(y,\beta),$ the function $g$ of $y$ depends on a parameter $\beta$, as shown in the graphs for $\beta=0, 6, 8, 10$, below. For values of $\beta$ 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 $\beta$, for $0 \le \beta \le 10$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $\beta=0$

    $\beta=8$

    $\beta=6$

    $\beta=10$

Thread navigation

Math 1241, Fall 2020