Math 519–About the 1st midterm

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Sample problems

  1. Find all solutions of $\dot x = x^2-x$, $x(0)=0$.
  2. Find all solutions of $\dot x = \sqrt{|x|}$, $x(0)=0$.
    1. Find a Lipschitz constant for the function $f(t, x) = x^2$ in the region $0\le t\le T$, $|x|\le M$.
    2. Find a Lipschitz constant for the function $f(t, x) = x(\sin(t)-x^2)$ in the region $0\le t\le 2\pi$, $|x|\le M$.
  4. Consider the function $f(x) = \sqrt{|x|}$.
    1. Is $f$ Lipschitz continuous in the region $|x|\le 1$?
    2. Let $\delta\in(0, 1)$ be some constant. Find a Lipschitz constant for the function $f$ in the region $\delta\le x\le 1$.
  5. Is the function $f(x) = x(1-\sqrt{|x|})$ Lipschitz continuous in the region $|x|\le 1$?
  6. Let $\delta\in(0, 1)$ be some constant. Find a Lipschitz constant for the function $f(x) = \{\delta +x^2\}^{1/4}$ in the region $|x|\le 1$.
  7. Does \[ f(t, x) = \begin{cases} x^2-x & 0\le t \le 1 \\ 2x & 1\le t\le 2 \end{cases} \] satisfy the Lipschitz condition in the existence and uniqueness theorem in the region $0\le t\le 2$, $|x|\le 1$?
  8. Let $x(t)$ and $y(t)$ be two solutions of \[ \dot x= f(t, x) \] on the interval $0\le t \le T$. Assume that $f$ satisfies the Lipschitz condition with Lipschitz constant $L$. Show that \[ |x(t) - y(t)| \le e^{+Lt} |x(0) - y(0)| \] and also that \[ |x(t) - y(t)| \ge e^{-Lt} |x(0) - y(0)| \] for $0\le t\le T$.
  9. For each of the following initial value problems decide if the solution exists for all $t\ge 0$, or only on a finite time interval $0\le t\lt T$.
    1.  $\dot x = x(1-x)$, $x(0) = \frac12$
    2.  $\dot x = -x(1-x)$, $x(0) = \frac12$
    3.  $\dot x = -x(1-x)$, $x(0) = 2$
    4.  $\dot x = x^2(1-x^5)$, $x(0) = \frac12$
    5.  $\dot x = x(2+\sin(t)-x)$, $x(0) = 1$
    Note that for some of these problems $x_*(t)=0$ and/or $x_\dagger(t)=1$ are a special solution. You can use this fact together with the uniqueness theorem to conclude that the solution you are looking at has to stay in some region of the $(t,x)$ plane.
  10. Find and solve the variational equation in the following cases.
    1. $\dot x = x(1-x)$, $x(0)=\alpha$, at the given solution $\bar x(t) = 1$.
    2. $\dot x = x(1-x^3)$, $x(0)=\alpha$, at the given solution $\bar x(t) = 1$.
    3. $\dot x = x(1-x)^3$, $x(0)=\alpha$, at the given solution $\bar x(t) = 1$.
    4. $\dot x = \alpha x$, $x(0)=1$, at the given solution $\bar x(t)=e^{t}$, $\alpha=1$.
    5. $\dot x = -\alpha x^2$, $x(0)=1$, at the given solution $\bar x(t)=1/(1+t)$, $\alpha=1$.
  11. A logistic equation with harvesting. A fish population in a large lake, left to itself, would grow according to $\dot x = x^2(1-x)$. In addition the natural birth and death in the population, fish are also being caught at a rate $h(t)$ (the “harvesting rate”). Thus the differential equation for the fish population which takes harvesting into account is \[ \dot x = x^2(1-x) - h(t). \] In the following questions you investigate the effect of a small amount of harvesting on a steady fish population.
    1. “Steady harvesting.” $\dot x = x^2(1-x) - \alpha$, $x(0) = 1$, at the given solution $\bar x(t) = 1$, and at $\alpha=0$.
    2. “Periodic harvesting.” $\dot x = x^2(1-x) - \alpha\sin \bigl(\frac{2\pi}{T} t\bigr)$, $x(0) = 1$, at the given solution $\bar x(t) = 1$, and at $\alpha=0$. Here $T\gt0$ is a positive constant (it may be simpler to abbreviate $k=2\pi/T$.)
  12. Draw the bifurcation diagrams for the differential equation $\dot x = f(a, x)$ for the following right hand sides $f$. In each case find all bifurcation points and determine if they are “standard bifurcations” or not.
    1. $f(a, x) = x^2 - a$
    2. $f(a, x) = x^2 - ax$
    3. $f(a, x) = x^3 - ax$
    4. $f(a, x) = x^4 - ax$
    5. $f(a, x) = x^3-ax+1$
    6. $f(a, x) = \sin(x)-ax$
    7. $f(a, x) = ax-x^3+x^5$ (see also problem 3, page 46/47).
    8. $f(x,a) = e^x-ax$
    9. $f(x,a) = e^{ax}-x^2$
    10. $f(x,a) = \ln(x) - ax$
  13. Assume that $(x_0,a_0)$ is a fold point in the bifurcation diagram for the differential equation $\dot x = f(x,a)$. Thus $f=f_x=0$ and $f_a\ne 0$, $f_{xx}\ne 0$ at $(x_0,a_0)$. Let the bifurcation set near $(x_0, a_0)$ be given by $a=a(x)$. Find an expression for $a'''(x_0)$ in terms $f$ and its derivatives at $(x_0,a_0)$.