MATH 417, Qualitative Theory of Differential Equations
- HWA-01, due Tue, Jan 16; exercises:
- Kong: 1.3(a), 1.7(c), 1.15,
- Strogatz: 2.2.8
- A full pot with hot water is standing in the sink.
Water is pouring into the pot at the rate of \(a\) gallons per minute and spills over the edge (at the same rate \(a\), of course) after being thoroughly mixed.
Denoting
by \(x(t)\) the temperature of the water in the pot at time \(t\), write the ordinary differential equation
for \(x(t)\).
- HWA-02, due Tue, Jan 23;
- Read and understand: Strogatz, Chapter 2;
- Exercises: Strogatz: 2.2.10, 2.3.4, 2.5.6, 2.7.3
- Let \(x(t)\) be the angle formed by the string of the pendulum with the downward vertical direction.
The angular acceleration of the pendulum is in direct proportion to the torque of the gravity relative to the
pivot. (a) Express this as a second order ODE for \(x(t)\). (b) Rewrite it as a system of first order ODE's.
- HWA-03, due Tue, Jan 30;
- Read: Strogatz, Chapter 3 and 4;
- Exercises: Strogatz: 2.5.3, 3.1.3, 3.2.4, 3.4.4, 3.7.3.
- Watch this and think.
- HWA-04, due Tue, Feb 6;
- Read: Strogatz, Chapter 4 and 5;
-
Exercises: Strogatz: 3.7.6, 3.4.11, 4.3.3, 5.1.3, 5.1.9(a-d).
- HWA-05, due Tue, Feb 13;
- Read: Strogatz, Chapter 5 and 6;
-
Exercises: Strogatz: 5.2.3, 5.2.13(a-b), 6.2.1, 6.3.1, 6.3.9(a-d).
- HWA-06, due Tue, Feb 20;
- Read: Strogatz, 6.1--6.4, 6.8, 7.0--7.3;
-
Exercises: Strogatz: 6.1.2, 6.8.5, 6.8.7, 7.3.3.
- Prepare at least one question.
- Midterm: Feb 22 in class.
- 4 problems from HWAs. (The calculation problems will have slightly different formulation.)
- 1 new problem (Comparable to the second problems on quizzes.)
- HWA-07, due Thu, Mar 1;
- Read: Strogatz, 7.0--7.2, 8.7;
-
Exercises: Strogatz: 7.2.2, 7.2.6a, 7.2.12, 7.2.18, 8.7.2.
- HWA-08, due Tue, Mar 13;
- Read: Strogatz, 6.5, 7.3;
-
Exercises: Strogatz: 6.5.1, 6.5.12, 6.5.19, 7.3.1, 7.3.4.
- (Optional) Watch "The science of sync" by Steven Strogatz (about 20 min)
- HWA-09, due Tue, Mar 20;
- Read: Strogatz, 6.6 and the pararagraph on Hamiltonian systems on page 189.
-
Exercises: Strogatz: 6.5.3, 6.5.9, 6.5.20, 6.6.1, 6.6.3.
- HWA-10, due Tue, Mar 27;
- Read: Strogatz, 6.6--6.7.
-
Exercises: Strogatz: 6.6.5, 6.6.10. 6.7.2(a--d), 6.8.4, 8.7.8 (Hint consider the flow on the cylinder \((x,t)\) where \(t\) is taken modulo \(T\)).
- HWA-11, due Tue, Apr 3;
- Read: Strogatz, 8.0--8.2.
-
Exercises: Strogatz: 8.1.2, 8.1.6, 8.1.12 (4 points), 8.2.9 a--c.
- Optional: Do 8.2.9d using the slope and direction fields applet of Darryl Nester.
- HWA-12, due Tue, Apr 10;
- Read: Strogatz, 8.2, 8.4, 9.1
-
Exercises: Strogatz: 8.1.5, 8.2.1, 8.2.5, 8.2.11,
-
A flow on the plane has only one fixed poitn at the origin and
\(P(x)= x+\frac{x\cdot(x-1)\cdot(x-2)}{(x+100)\cdot(x+101)\cdot(x+102)}\)
is its Poincare map which is defined for the positive \(x\)-axis.
Classify the fixed point; how many cycles the system has; classify each. Explain why there is no more cycles. Do you think it is possible for this system to have an orbit which does not cross the \(x\)-axis?
- HWA-13, due Tue, Apr 17;
- Read: Strogatz, 9.2--9.5, 12.1, 12.3
-
Exercises: Strogatz: 8.2.13, 8.4.2, 9.2.2, 9.2.4, 12.1.7.
- HWA-14, due Tue, Apr 24;
- Read: Strogatz, 11.0--11.4
-
Exercises: Strogatz: 9.3.8, 11.3.2,
11.3.5, 11.3.8, 11.4.6(a--c), 12.1.5 [part (e) is optional].
- Final exam: Tue, May 1, 104 Thomas building, 14:30--16:20.
- 6 problems from HWAs. (The calculation problems will have slightly different formulation.)
- 2 new problems (Comparable to the second problems on quizzes.)
- Cheat sheet: you can bring a sheet of paper one side, with diagrams and formulas, no words allowed.