Algebraic Solutions of Differential Equations
10t
dependent elements of the image, it will follows that the target has
dimension
precisely
two, and that (6.8.6.0) is surjective, and hence an
isomorphism (source and target having the same dimension). By (6.8.4),
the image contains the classes
dx
x dx
(6.8.6.1)
ye ,
yt
whence it suffices to prove that they are linearly independent in
P()~(E))H~R(X(n;
a, b, c)/C(2)).
Suppose the contrary. Then there
exist (by (6.8.1.8))
~, fl ~ C(it),
not both zero
(6.8.6.2)
R (x) e B = C (it) Ix] [
1/x (x 
1) (x  it)]
such that
(6.8.6.3)
d ~[ R(X)]ye ]
=(~+flx)
7dx
in
~"~A(n;a,b,c)/C()O
This implies that on the open set where
R(x)
is invertible, we have
dR(x)
dy
(o~ + fl x)
(6.8.6.4)
(
=
y
or, equivalently
(6.8.6.5)
/ a/n
bin
c/n \
(~ + fl x) dx
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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