Algebraic Solutions of Differential Equations 10t dependent elements of the image, it will follows that the target has dimension precisely two, and that (184.108.40.206) 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 (220.127.116.11) 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 (18.104.22.168)) ~, fl ~ C(it), not both zero (22.214.171.124) R (x) e B = C (it) Ix] [ 1/x (x - 1) (x - it)] such that (126.96.36.199) d ~[ R(X)]ye ] =(~+flx) -7-dx 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) (188.8.131.52) ( = y or, equivalently (184.108.40.206) / 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.