Asymptotic theory for third-Order differential Equations

Abstract

Abstract In this Ph.D. thesis we investigate the form of three linearly independent solutions of two types of the third-order linear differential equations as x + @. Asymptotic theory will be developed for these two differential equations with large coefficients that are like powers of the independent variable. The first type is the differential equation {s(s y') ' ]' + qy' + ry = 0, (asx and we shall establish the asymptotic formula Y, (x)~ s (x) r (x) exp ["KE) (t) - ux 6 a (t) ]at as x + @, for three solutions of the differential equation (E1) subject to certain conditions on s(x), q(x) and r(x) as stated in section 4, Theorem 4.1. The second type is the differential equation {s(sy') ']' - (py')' - ry = 0, and for this differential equation we shall consider two cases, case (1) is given in section 9, Theorem 9.1, and under this Theorem we obtain three linearly independent solutions y(x) of the differential equation (E2) such that ¥k(x)v [p(x)r(x))-t exp1[[=1(r/p)\(t)-s 段 (t)]at} as x + @ (k = 1,2) (a ≤ x and ) = 01 p (x) s (x) r (x) exp P(t)dt + X SI(t)dt p as x + @. Also case (2) of the differential equation (E2) is given in Section 13, Theorem 13.1, and under this theorem we obtain three linearly independent solutions y (x) of the differential equation (E2) such that Y1 (x) = of (p(x) r(x) f*), Y2 (x) (p(x)r (x) )" exp Y3 (x) = o (p(x) r (x) ) " exp as x + @ 2ơ a x P(1+¢)dt], S 3 a s as x + @ t+ (1+u) f Bat as x + @.