By Choudhary P.

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**Example text**

An important case to begin with is that where a ( t ) is close to a constant, which is to say, the equation has the form u” + (1 +f(t)). 2) where f is small in some sense. 1) with the aid of a simple basic transformation due to Liouville. * Exercises + + 1. Consider the equation u” (1 Ef(t))u = 0, where E is a parameter. Show that we can obtain a formal solution of the form u = uo E U ~ e2u2 + , where + + u; + u, = 0, u; + u1 +fu, = 0, and so on. We will examine the validity of perturbation expansions of this nature in Chapter 3.

1) in terms of a first. u2 Exercises s" 1. 20. THE SECOND SOLUTION 2. Simplify this in the case where f = sin t/l/F. 3. for large t, and obtain corresponding expansions for u1 and u2 . Are the expansions convergent ? + 4. What is the eventual behavior of the solution of v' v 2 - 1 -f(t) = 0 with v(0) > 1 iff + 0 as t + 03 ? Hint: Consider the behavior of v in the (v, t)-plane. 20. 1) 0 appears to have a solution that approaches -1 as t + GO, provided that f -+ 0 as t -+ 03. We can establish this indirectly as above by using the properties of u p .

T h e determination of when u 2 ( T )# 0 is related to questions of characteristic values, which we shall discuss iubsequently. Exercises + 1. Solve u" u = 0, u(0) = c, , u ( T ) = c2 . Discuss the condition on T to ensure existence and uniqueness. 8. CONNECTION WITH CALCULUS OF VARIATIONS 11 u’( T )= c 2 , and when u(0) = cl, u( 7’) = c2 . Discuss the question of existence and uniqueness. 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.