### 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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### Material on Test 1

Test 1 is Wednesday Oct. 10. The review day is Monday Oct. 8. In the Schaeffer-Cain text, the test will cover what we have done in Chapters 1 and 2.

Things you should be able to do:
• Solve initial value problems in one dimension using separability and linearity. (Sec. 1.8, problems 3c, 3d.)
• For systems dx/dt=f(x,y), dy/dt=g(x,y), derive a differential equation for the orbits and solve where possible. Also, check that a proposed first integral really is one. (Sec. 1.8, problem 5.)
• For systems dx/dt=f(x), dy/dt=g(x,y), solve initial value problems by first solving the first equation, then substituting into the second. (Sec. 1.8, problem 7; Homework 5, problem 2.)
• Sketch phase portraits for x'=f(x), x in R, and for x''+f(x)=0. (Homework 2 problems 1-3).
• Use the basic theory of linear differential equations from Teschl, sections 3.1-3.2. (Homework 2, problem 4; Homework 4, problem 5; Homework 5, problem 2.)
• Do problems involving the operator norm and the matrix exponential. (Homework 3; Homework 4, problems 2, 3.)
• Do problems that involve differentiating matrix-valued and vector-valued equations (Homework 4, problem 3; Homework 5, problem 1.)
• For x'=Ax, find the Jordan form J, e^{Jt}, and the change of basis matrix P. (Homework 4, problem 1.)
• Be able to perform change of coordinates for linear differential equations (Homework 4, problems 2, 4).
• For x'=Ax, x in R^2, use eigenvalues and eigenvectors to sketch the phase portrait (Homework 5, problem 4).
• Be able to deal with complex-valued matrices (Homework 5, problem 3.)