16 1. Differential Equations and Their Solutions σp(t) = ektp i.e., in this case the flow map φt is just multiplication by ekt. • Example 1–3. Harmonic Oscillator. If we start from the Harmonic Oscillator Equation, d2x dt2 = −x, and use the trick above to rewrite this second-order equation as a first-order system, we end up with the linear system in R2: dx dt = −y, dy dt = x. In this case the maximal solution curve σ(x 0 ,y0) (t) can again be given explicitly, namely σ(x 0 ,y0) (t) = (x0 cos(t)−y0 sin(t),x0 sin(t)+y0 cos(t)), so that now φt is rotation in the plane through an angle t. It is interesting to observe that this can be considered a special case of (a slightly generalized form of) the preceding example. Namely, if we identify R2 with the complex plane C in the standard way (i.e., a+ib := (a, b)) and write z = (x, y) = x + iy, z0 = (x0,y0) = x0 + iy0, then since iz = i(x + iy) = −y + ix = (−y, x), we can rewrite the above first- order system as dz dt = iz, which has the solution z(t) = eitz0. Of course, multiplication by eit is just rotation through an angle t. It is very useful to have conditions on a vector field V that will guar- antee its completeness. Exercise 1–12. Show that σp(t) − p≤ t 0 V (σp(t)) dt. Use this and the No Bounded Escape Theorem to show that dx dt = V (x) is complete provided that V is bounded (i.e., sup x∈ Rn V (x) ∞). Exercise 1–13. A vector field V may be complete even if it is not bounded, provided that it doesn’t “grow too fast”. Let B(r) = sup x r V (x) . Show that if ∞ 1 dr B(r) = ∞, then V is complete. Hint: How long does it take σp(t) to get outside a ball of radius R? Exercise 1–14. If a vector field is not complete, then given any positive , there exist points p where either α(p) − or ω(p) . 1.2. Euler’s Method Only a few rather special initial value problems can be solved in closed form using standard elementary functions. For the general case it is

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2009 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.