Question

1. Write the forms of the series expansion about the regular singular point x=0 for two linearly independent solutions to the following differential equation (do not compute the coefficients in the expansions): zºy'(x) - xy'(x) + (1 - x)y(x) = 0. (1) 2. Does the following system of equations have an unique solution? Explain your reason for the answer (do not find the solution). 2x 1 + 4x2 + x3 = 8 2x + 4.62 = 6 -401 - 8x2 + x3 = -10

Answer 1

no ? M-2 na 2 M-71 no Иго 12 n2 MO o 22 y"100) -& y'ex) + ( =*) y(x) = 0 - 0 Let y (x) = Ca xh Sy's nen that > yes = SŠ mavcnt** ...OS -m-1) (vxº- £ 64*** Conxa ŠCom no > n(n-d.com"-16, x' n ong it cox't 't de la conta de Can X4 20 > { min-) cn -Cu + Cn --+34 760-650 1. Win-1) en hently. Can 2014072 > (nea-) – +1) (n = (n =0,6622 » (1²_n-_nt) in = Chri , &n32 =) (n²-2n+1) (n = Cal th>2 = (n = Cannot sense Hence the solution is of the form Yer) er n ene 24 + 60+4,2

* 2x+4*2 + x3 = 8 28, +4% =6 . . -444-822 + z = -10 Rewrite in the matrin forum AX=B [2 4 11,7 ſ 2 4 0 4 -8 1 31 2 4 118 Thee Qugareeted matrix is se 41B) 240 1-4-811-0 Guie is avel = 0.13) 12.4.118 1 Ro Rz -R, Too- 11- 21 = Loo 3 16 | Rrez Riz +2R, Av og doo 4 2 o 161 Rp) Rathia 1/2 Rry Horro Rrz Ryt 3kra [13:13 Ray-ER ? Looolo moeuns. there the rank of [918] = 2 f the stank of Im] =2 Le) thee sank of [AB] = the rank of [m] = 2 < the Bomber of unknowns. . Given system have infinite Ho of solutions. de given system of equation have no anique solution .