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QUESTION:

We consider differential equations with singular points at to. (25 info\" : (t t0)b(t) : C(m _ 0 where Mt) and CE) are analytic functions in |t t0| 0. This singular point is called a regular singular point. For, example the differential equation t337II+(COSt*1)I'+tC 2 has a regular singular point at t = 0. The dierential equation t2(t 1)2:r\"+ sintr' + 90: 1):c : 0 has a regular singular point at t = 0, but the singular point t = 1 is not regular. A change of variables t I> t to translate the singular to 0. We will study equations of the form i521" + tb(t)a:' + C(tJm = 0 with m 00 b(t) = 2W} C(t) = Zane\" 1:0 1:0 on |t| 0 1. If r1 r; is not an integer then the differential equation has two linearly independent solutions 00 DO 931(t) = tn Zaiti} 332(t) = tn Edit-1 i=0 i=0 where the sums converge on Itl (-1)' (2j + 1)! For r = 0, aj aj+1= (23 + 1) (23 + 2) a1 = ao , a2 = - al ao . .. ,aj = (-1)j- do 12 24" (2j ) !) X2 ( t) = (-1 )j- it ( 2j ) ! 2 tax" +tx' - tx = 0 bo = 1, Co = 0, r2 = 0. One root, r = 0. ElGi - Da,t' + Ella,t - Za,titi = 0 [ (+ 1 ) ()ajitit + [ (+ 1)ajitity Lajtiti = 0 aj aj+1= (3 + 1)2 a1 = a0, a2 = do a2 ao do 4 ' 36 . . .,aj ( 7!) 2 ti x1 (t ) = > 2 (!)2, X 2 (t ) = (Int) x (t ) + t Ed, ti The coefficients of the second solution can be determined using the same procedureUse Frobenius method of Lecture 11, to find the first three non-zero terms of the solutions: tha" + mac' + (ott) ac = 0 Upload Choose a File