Question

please show how to do without sigma notation i.e. in table form as pictured

oal.c beu 6x. Use powe series to find the general solution of (x2+1)y"-4y y(x)= a + rD a,x ax y' a, +2a,x 3a, x2 4a, x + y =2a, + 6a,x + 12α,x. 20ας x -1)a x"2 -2 -2xy = -2ag x- 2α,x 2a,x + + 0x + Ox2 Ox Ignoring the sigma notation for the moment, adding the terms of the last two rows and equating coefficients of powers ofx with those of the RHS of the ODE (all 0 in this case) leads to: 2a,=0 a, -0 1 6a,-2a-0 3 1 12α, -2a, 0 a 6 a 20a, -2a, 0 0 Hence, the series solution can be written as: y(x) = a, + a,x+a,x+a,r+ I 1 =a+ax+a or 6 1 y(x) = a (1 +x +) - 3 .). (x Notice that there are two linearly independent solutions, each multiplied by an arbitrary

Answer 1

() Ler Sa be ya, eMay Put theu val in 0 - (a,+ at 0g ?) - - 4 ag 29, Compng Cont I 292 1294-p o12 2 fas t 20 4p -Yag =o 2oa-203 1 , (243 30 honu t Sot