# 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. 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 $q(0)=0$.

3. Consider the dynamical system \begin{align*} \diff{ x }{t} &= f(x) \end{align*} where the function $f$ is graphed to the right.
1. Find all equilibria.
2. For each equilibrium, determine its stability by sketching the solution for initial conditions just above and below the equilibrium.

4. 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 $y(0)=2.5$.

5. Estimate the solution of the differential equation \begin{align*} \diff{ x }{t} &= - 2 \ln{\left (1.4 x^{2} \right )}\\ x(0) &= -0.1, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.5$ to estimate $x(1.5)$.

6. For the dynamical system $\diff{ x }{t} = f(x,\beta),$ the function $f$ of $x$ depends on a parameter $\beta$, as shown in the graphs for $\beta=-6, -3, -2, -1$, 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 $-6 \le \beta \le -1$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

$\beta=-6$

$\beta=-2$

$\beta=-3$

$\beta=-1$

7. For the dynamical system $z'(t) = f(z, c)$, where the function $f$ of $z$ 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. $c= -8$
2. $c= -1$
3. $c= 3$
2. Identify any bifurcation points.

8. Consider the dynamical system \begin{align*} \diff{ z }{t} &= -3 \left(z + 7\right) \left(z + 5\right) \left(z - 2\right)\\ z(0) & = z_0, \end{align*} where $z_0$ 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 $z_0 = -9.1, -6.0, -3.6, 5.2$.

9. Consider the differential equation \begin{align*} \diff{ z }{ t } &= -8 z. \end{align*}
1. What is the general solution?
2. What is the specific solution for the initial condition $z(0) = -6.3$?