By Katz N.M.

By Katz N.M.

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Example text

5. 5#3b that this "curve" is a weak upper fence that meets the line x = t. A solution starting at a point on the x-axis will stay below this fence until it reaches the line x = t, after which it will stay below one of the hyperbola fences described above. Therefore no solutions can escape the funnels and antifunnels described above. X " ....... / / ' ........ / / / ....... ,.. __ ,,,, ....... 6. x' = sin tx. Slopes marked on a grid. It can also be shown that for positive k, the solutions to this differential equation have 2k maxima ( k on each side of the x-axis) and that nonexceptional solutions in the first quadrant lie in a funnel, which we shall call 40 1.

3. Consider cos 2 x 1 x =1+At2=f(t,x), for A>O, an equation we will meet in Volume III, in the analysis of Bessel functions for a vibrating membrane. What can you say about the solutions as t--+ oo? There are two good equations {because you can solve them and use them for fences) with which to compare the given differential equation. 8, with the slope marks for solutions to the differential equation as indicated according to the inequality. 8. Funnel for x 1 = 1 + A(cos 2 x)jt2. (Actually, according to the inequality, a(t) and f3(t) are weak fences, so we are a bit ahead of ourselves with this example, but this is a very good example of where we will want to use solutions to similar equations 42 1.

Some solutions fly up, some fall down; exactly one exceptional solution does neither and separates the other two behaviors. 5. 1. 7 is of particular interest because although it looks utterly simple, there are no formulas in terms of elementary functions, or even in terms of integrals of elementary functions, for the solutions. A proof of this surprising fact is certainly not easy, and involves a branch of higher mathematics called Galois theory; we do not provide further details at this level.