Question

h Bessel equation of order p is ty" + ty + (t? - p2 y = 0. In this problem assume that p= 2. a) Show that y1 = sint/Vt and y2 = cost/vt are linearly independent solutions for 0 <t<o. b) Use the result from part (a), and the preamble in Exercise 3, to find the general solution of ty" + ty' + (t2 - 1/4)y = 3/2 cost.

Answer 1

of order p is Given Bessel equation ry" + ty + (4² p² y = 0 Putting P = ty" + ty + ( 12 4 )y=0 V Vt Y2 = Cost for old coo. (as to show that I sint and are linearly independent solutions ist we need to show that I and solutions of eq. O, then we need to of and Y2 are linearly independent .: y = sint V+ Y2 are show that for OltLoo - y = + 1/2. Sint so that y' = - 2 + 3/2 sint + + 1/2. Cos (4) and g = 2 + 1/2 sint - + 3/2 (st - + 12. Sint Now satisfying y G y inneg. we get L=1² ( 3 4 5 1 2 Sint - +312 cost - #12 sint ] ++ [ - 2 + 12 sint + + "2. (st] + (x2 - 14 ) + 12 sint = 3 + 1/2 sint – + 12 Est - +312 ont - 1 t 1/2 sint + + "2 est + + 3/2 sint - + 12 sint = + 12 sin (4) 24- 24 = + 12 sin (4) [ 8-17 = 1/2 fint. (o) = 0 = R. Hts. Thus L.H.S. = RH-S. Thus y sint is a solution to eq. O Vt Maw : Y = Cost y = + V2 Cost Vt so that y = - 12 +312 Cost - tl2 sont and g = 2 Cost. t 512 + + 3/2 sint - #12. est -

on How satisfying values of Y ' s Lits of eq. O, we get 1.His = 2 1 2 1 5/2 cast + 312 hint - tv2 eart) + + 1 - 2 + 1 cast - - 12 hint) + (12 4 ) . 2 est = 3 + 42 cast + Alla fine - Baut - 1 + 12 Cost - 2 sint + 3/2 cost - 1 4 - la cost - + -] =.654 [1-6] = tf2 cost [0] = 0 =R-H.S. Thus LH-S. = R.H.SI Y = Cost This is also a solution V: to eq.O. Maw Wroustian + 12 sint wly, Y₂) = + 112 cost 1 2 + 3/2 sint + + Chest Mast - 1/2 fint =xVsn-tex Wats-Vesn] - #126st (- 4 3/2 sint + + 12 cast] a sint cost t sinkt + 1 & sint Cost - shut = -Al sin?t t asht -bit - Casey an?4 468°41] Wly, Y = -1 .: Sint 0 + GR²0 = 1 OLA 200 & W(GY) to for Thas Wly, Ys) to . I and Y are Hence y, and ų are linearly for Octcool linearly independent. independent solutions

(b) using result from general solution of part (as to find the ty" +ty + (+24) 4 = 1312 Cost from part (as it is proved that y and I are linearly independent & y + xy + (42 +) y = 0 solutions of eq. I lot = y + y where G, G are arbitrary, Constants. Vt C.F. = a sint & C2 Cast How let particular integral be P.I. - y u + L₂ U2 - where 4 =- | Yr . glt) de -dt I wly .4) and - 1 y. gets - dt Given equation is ty" + xy + (x - 1) y = 4312 cost dividing by 12, we get 4 + (+ 4) = Cost which is of the form y" + 4 8 trong +S(t). Y = g(t) where JG)= Cost → 4 = | Cost/4 J olt + 4 = 1 Casht at 554 = f ( 4 (624) 4 = { 1 + 65 (20) at Cos(24) = 2 Cos? H11

Mow U2 = ( y.gt). I W(Y,Z) sint. est - dt → 42 = -) sint (ast at > U2 = -f Sin (24) at 2 sint Gst- &in (24) 4 cs(24) How putting w Y , L , 4 , U2 in g. . we get Pisa Sant 4 in.(20+ Cast. Gx(24) 4- This general solution will be you = C.F. +P.I. c) gommate sangue carte