### MA 532-001 Fall 2012 Ordinary Differential Equations I

Mathematics

Ordinary Differential Equations by Schaeffer and Cain | Feedback on text
Ordinary Differential Equations and Dynamical Systems by Gerald Teschl
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### Homework Assignments

#### Assigned Tuesday, Aug. 21

Homework 1: Sec. 1.8, problems 3c, 3d, 5, 6, 7. Due Friday Aug. 31. For instructions on how to submit, go here and scroll down to Homework.

More problems may be added later this week.

Problem 3d: What happens to your solution when x (not t) approaches 0?
Problem 5 refers to equation 1.39 in the text. Warning: equation 1.47 is upside down (numerator and denominator should be exchanged). You should derive the equation L(x,y)=c by solving 1.47 by separation of variables. Then you should check your answer by checking that if (x(t),y(t)) is any solution, then (d/dt)L(x(t),y(t))=0.
Problem 6a: You can do this by showing that x_{partic}(t) - x(t) satisfies the homogeneous equation.
Problem 6d: You can use Maple or something similar to make the graph.

#### Assigned Sunday, Sep. 9 and Monday, Sep. 10

Homework 3: Prove Lemma 2.2.3 parts (i)-(iii) and Lemma 2.2.4. Note that the statement of Lemma 2.2.4 continues onto the next page. Also: Problem 5(a) in Sec. 2.6.
Due Friday Sep. 14.

In Lemma 2.2.3, ||A|| is the operator norm given by (2.12), in which you should interpret |x| as any norm on R^d (so it satisfies the properties listed in Lemma 2.2.2).
In Lemma 2.2.4, |x| is the Euclidean norm, and ||A|| is the operator norm.
In Problem 5(a), don't use Jordan form. Follow the hint, and compute the exponential as an infinite series. You should recognize the sum.