In solving a differential equation by Frobenius method of series solution about x= 0, we obtain the following indicial equation and recurrence relation: (27-1)=0 and Cn+1= 6n +r+3 n where n= 0,1,2,3,... Find the solution corresponding to the larger root of the indicial equation. Include the first three nonzero terms, use Co=1 OAY =1+ 133 OBy =x-1/2| 43 +.. Ocy=x1/2 x+ +1 133 +: +.. 19

3) Find the first five terms, ao, a₁, a2, a3, a4, in the series solution at x = 0 of the initial value problem (IVP) (x - 1)²y" + xy + y = 0, y(0) = 0, y'(0) = 1.

Consider a series solutions about the point x = 0 for the ODE: y" - xy + xy = 0 (i) Find a recurrence relationship for the coefficients of the series. (ii) Using (i) to evaluate the first four non-zero terms of each of the two linearly independent solutions and state the general solution for y(x).

The Legendre’s differential equation is  (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x). Consider a series solution about the point x = 0 of the form provided, where k is a constant and the an are coefficients that need to be determined. Show that the recurrence relation is given by an+2 (n + k + 2)(n + k + 1) = an [(n + k)(n + k + 1) − l(l + 1)].