Use the power series method to solve the initial value problems below (1.1) (1.2) (1-x²)y" - 6xy' - 4y = 0; with y(0) = 1; and y'(0) = 1 (x²+1)y" - 6xy' +12y = 0; y(0) = 1, y'(0) = 1.

Consider the differential equation x^2y" + x (1−x)y' − xy = 0. a. Show that X0=0 is a regular singular point. b. Determine the indicial equation, the recurrence relation, and the roots of the indicial equation. c. Find the series solution for x>0 corresponding to the larger root. d. Find the series solution corresponding to the smaller root by following the procedure outlined in section 5.4 and demonstrated in section 5.7.

Find the power series solution of the given initial value problem y''-xy=0 y(0)=1 y'(0)=-1

Consider the following differential equation to be solved using a power series. y'' + xy = 0 Using the substitution  y =  ∞ cnxn   n = 0 ,  find an expression for  ck + 2  in terms of  ck − 1  for  k = 1, 2, 3  . ck + 2 =  Ck−1​(k+2)(k+1)​