Homework for Math 6410 §1, Fall 2012 Andrejs Treibergs, Instructor August 28, 2012 Our main text this semester is Lawrence Perko, Differential Equations and Dynamical Systems, 3rd. ed., Springer, 1991. Please read the relevant sections in the text as well as any cited reference. Each problem is due three class days after its assignment, or on Monday, Dec. 10, whichever comes first. 1. [Aug. 20.] Compute a Phase Portrait using a Computer Algebra System. This exercise asks you to figure out how to make a computer algebra system draw a phase portrait. For many of you this will already be familiar. See, e.g., the Maple worksheet from today’s lecture http : //www.math.utah.edu/~treiberg/M6412eg1.mws http : //www.math.utah.edu/~treiberg/M6412eg1.pdf or my lab notes from Math 2280, http : //www.math.utah.edu/~treiberg/M2282L4.mws. Choose an autonomous system in the plane with at least two rest points such that one of the rest points is a saddle and another is a source or sink. Explain why your system satisfies this. (Everyone in class should have a different ODE.) Using your favorite computer algebra system, e.g., Maple or Matlab, plot the phase portrait indicating the background vector field and enough integral curves to show the topological character of the flow. You should include trajectories that indicate the stable and unstable directions at the saddles, trajectories at the all rest points including any that connect the nodes, as well as any seperatrices. 2. [Aug. 22.] Continuous Dependence for Constant Coefficient Linear Systems. Let A be an n × n real matrix. For every x0 ? R n, let ?(t; x0) denote the solution of the IVP ? ??? ??? dx dt = Ax, x(0) = x0. Note that ?(t; x0) is defined for all t ? R. For fixed t ? R show that limy?x0 ?(t; y) = ?(t, x0). 1 3. [Aug. 24.] Real Canonical Form. Let A be a real 2 × 2 matrix whose eigenvalues are a ± bi where a, b ? R such that b 6= 0. Show that there is a real matrix Q so that Q -1AQ = a - b b a . Use this fact to solve the system x 0 = -13x - 10y y 0 = 20x + 15y 4. [Aug. 27.] Jordan Form. Find the generalized eigenvectors, the Jordan form and the general solution y? = ? ??????? XXXXXXXXXX8 ? ??????? y. 5. [Jan. 29.] Just Multiply by t. Consider the n-th order constant coefficient linear homogeneous scalar equation x (n) + an-1x (n-1) + · · · + a1x 0 + a0x = 0 where ai are complex constants. Convert to a first order differential system x 0 = Ax. Show that the geometric multiplicity of every eigenvalue of A is one. Show that a basis of solutions is {t k exp(µit)} where i = 1, . . . , s correspond to the distinct eigenvalues µi and 0 = k < mi where mi is the algebraic multiplicity of µi . [cf., Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Amer. Math. Soc., 2012, p.68.] 2