(ALNEELAIN UNIVERSITY, 2003) MOHAMMED ABD AL—BAcI MOHAMMED
ABSTRACT
Lie's group theory of diflerential equations was initiated by the
Norwegian mathematician Marius Sophus Lie (1842-1899). Today, this
area of research is actively engaged.
In chapter one of this thesis, we give the fundamental concepts of
the one-parameter Lie group of transformations, and it also contains the
main theorems and definitions.
In chapter two, we apply the Lie's theory to the following second
order ODE's
x:_v3+xy1-1=0, (1)
xiv; —y,: —l = 0. (2)
_v: —y1 —£=O. (3)
y
-t:(1”:+,t:_v]: — Zxyl + 2 = 0 . (4)
k
d y
h . = i . k = 1.2.
w ere y_ LN
In this chapter. we obtain the following results
(i) The symmetry groups of(1), (2). (3). and (4).
(ii) Reduce the order of (1). (2), (3), and (4) to the first order ordinary
differential equations.
(111) The general solution of (1). (2), (3), and (4).
In chapter three, we considered the non-linear third order ODE
y;+Zyy:-_v|2=O. (5)where _vk
which is k
In this cha
(i)
(ii) Red
equ
(m) The
(iv) Ne
The symmetry groups of (0)
w
__-Q _ ,___ _
k
— d f k -1 v 3
dx
nown as Goldstein equation [7].
pter, we obtain the following results
uce the order of (5) to the first order ordinary diflerential
ation.
invariant solutions of (5).
solutions from known solutions for (5).
IV
