Weekly problems

Problems marked "Week n" relate to material from week n ¨C 1. You should strive to solve these in week number n.

Week 4

1. What is the order of the following ODEs? Reduce them to systems of first order ODEs.
   1. \(\ddot{x} + 2\dot{x} + \sin(x) = 0\)
   2. \(\dddot{x} + 3\dot{x}\ddot{x} - x^3 = 0\)
   3. \(t^{-2}\dddot{x} = x^2\dot{x} + \ddot{x}\log|\dot{x}|\)
2. Solve the following initial value problems:
   1. \(\dot{x} = -\sqrt{\alpha^2-x^2},\ x(0)=\frac{\alpha}{6}\) for some \(\alpha>0\)
   2. \(\dot{x} = (1+x^2)e^{-t},\ x(0)=0\)
3. Solve the following initial value problems and describe the qualitative behaviour of the solution.
   1. \(\ddot{x} + x = 0,\ x(0)=\dot{x}(0) = 1\)
   2. \(\ddot{x} - \dot{x}-2x = 0, \ x(0)=3,\ \dot{x}(0)=0\)
   3. \(\ddot{x} + 2\dot{x} + 5x = 0,\ x(0)=3,\ x'(0)=6\)
4. Consider the initial value problem \(\ddot{x} - \dot{x}-2x = 0, \ x(0)=1,\ \dot{x}(0)=-1\). 
   1. Find the solution of the problem. Describe its qualitative behaviour.
   2. What do you think will happen if we commit a small error, say, in the initial data? Will the qualitative behaviour of the solution remain the same? 
      (We will investigate instabilities closer when we discuss numerical approximations.)
5. Consider the logistic equation with harvesting term:
   \(\dot{x} = x(1-x) - h\)
   for some \(h\geq0\). This is a model for the population of an animal where a fixed number h is harvested at regular intervals. Find the solution with initial data \(x(0) = x_0\). For different values of h, plot the solution for a few choices of \(x_0\) and describe its qualitative properties.

Week 5

• Exercises in Section 2.1: 5, 6, 8, 9, 10
• Exercises in Section 2.2: 1 (do as many of these as you can), 2, 4 (prioritize this one!)
• Exercises in Section 2.3: 1, 2

Week 6¨C7

• Exercises in Section 3.1: 1, 2, 3
• Consider the equation \(\dot{x} = -x^3,\ x(0)=x_0\).
  • Verify that the velocity field \(F(x,t)=-x^3\) is locally, but not globally Lipschitz continuous.
  • Conclude that a solution exists locally, but possibly not globally.
  • By inspecting the ODE, show that solutions are bounded for all t, and conclude that a solution exists for all t.
• Consider the equation \(\dot{x}=ax,\ x(0)=x_0\) for some numbers \(x_0,a\in\mathbb{R}\).
  • Compute the solution of the ODE.
  • Compute the first five iterations of the Picard iterations. Recall that these are defined by \(x_1\equiv x_0,\ x_{n+1}(t)=x_0+\int_0^t F(x_n(s),s)\,ds\), where F is the right-hand side of the ODE.
  • Can you find a formula for \(x_n(t)\)? Does \(x_n\to x\)?

Week 8

• Exercises in Section 4.1: 1, 3
• Exercises in Section 4.2: 1, 2, 3
• Exercises in Section 4.3: 1, 2, 5

Week 9

• Exercises in Section 4.4: 1, 2, 3, 4

Week 10

• Exercises in Section 4.5: 1, 3, 4, 8, 11
• Exercises in Section 4.6: 3

Week 11

• Exercises in Section 5.1: 1, 2
• Exercises in Section 5.3: 1, 3, 5

Week 12

• Exercises in Section 5.5: 1¨C4
• Determine whether each of the following systems are (i) gradient systems, (ii) Hamiltonian systems, or (iii) neither. In case (i) and (ii) use this information to draw a phase portrait. (Use a computer to draw a contour plot, if needed ¨C but try to do it by hand first.)
  • \(\dot x = -2xy^2,\ \dot y = -2x^2y\)
  • \(\dot x = y-3,\ \dot y = 2-x\)
  • \(\dot x = 2xy - y^2, \ \dot y = x^2y\)
  • \(\dot x = x^2y,\ \dot y = -xy^2\)
  • \(\dot x = x^2 - 1\)
  • \(\dot x = -\cos(x)\cos(y),\ \dot y = -\sin(x)\sin(y)\)
• Consider the linear system \(\dot u = Au\) for \(A\in\mathbb{R}^{2\times 2} \). Determine for what A this system is (i) a gradient system or (ii) a Hamiltonian system. Can the system be both?
• Problem 6.1.1

Week 13

• Problems in Section 6.2: 4, 5, 6
• Problems 8.1 and 8.2 in my lecture notes

Week 17

Problems 9.1¨C9.3 in my lecture notes.