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2. Solve u” - au =0, u(0) = c,, 3. Consider the equation u“ + p(t)u’ + q(t)u = 0 subject to Let u, , u2 be two linearly independent solutions of the equation. Show that the question of existence and uniqueness of a solution subject to the foregoing conditions is dependent upon the nonvanishing of the determinant 4. Show that 5. Show that the domain of integration may be taken to be 0 < t, < t , < T . + 6 . Using this result, discuss the problem of solving u” u = 0 subject T u ( t ) elit dt = b, , J i u ( t )eA21dt = b, .

It is a particular example of an asymptotic series. These series of paramount importance were introduced into analysis in a systematic fashion by PoincarC and Stieltjes. Let us now give a formal definition. We say that a series S ( t ) = a, a +2 + -a$ + ... 5) is an asymptotic series for a functionf(t) as t -+ co if the coefficients are obtained in the following fashion: a, = limf(t), a, = l i m t ( f ( t )- a,), an = l i m t n [ j ( t )- a, - ... - an-1 t-tm t+m t+m 31. 8) we mean that e - t f ( t ) possesses the asymptotic series Cn=, a,tpn.

7. Let 7 be any third term. Th en the quotient (atbuCu’ - dtb-1uc+1)/7 tends to a limit as t + CO. There are now two possibilities: (a) There is a third term whose order is equal to that of the difference between the two principal terms. (b) There is no such third term. 8. Consider the first possibility. 2). 9. Complete the analysis for part (b) of Exercise 7. More detailed results can be obtained both for the solutions of polynomial equations of the form p ( t , u , u’)= 0 and for equations of the form u” = p(u, t)/q(u,t ) .

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Methods of Nonlinear Analysis - Volume 1 by Richard Bellman

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