Dr. Katz DEq Homework Solutions 101

Dr. Katz DEq Homework Solutions 101 - A lgebraic Solutions...

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Algebraic Solutions of Differential Equations 10t dependent elements of the image, it will follows that the target has dimension precisely two, and that ( 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 ( 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 ( ~, fl ~ C(it), not both zero ( R (x) e B = C (it) Ix] [ 1/x (x - 1) (x - it)] such that ( 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) ( ( = y or, equivalently ( / 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.

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