1. Consider

x² y" + (x²+x) y' +(2x-1) y = 0 ,(1-1)

a)Can you use the Frobenius method to find a general solution?

b)Find the indicial equation based on (38) on page 210 and calculate the roots r₁ and r₂ , (r₁?r₂) (Hint, use the l'Hopital's rule, if necessary) ;

c)Use the recursive approach, verify the indicial equation;

d)Based on theorem 4.3.1, can you first use r₂ to have the general solution? And explain your judgement.

e)Find the general solution of y₁ with r₁ and calculate the coefficient up to c₄ and also find the general expression of the recursion formula, (recursion formula for y₁)

f)Find the general solution of y₂ based on theorem 4.3.1. (Hint, set d₂ = 0)

theorem 4.3.1 and No.(38) is in the following picture

THEOREM 4.3.1 Regular Singular Point; Frobenius Solution Let :1: = 0 be a regular singular point of the differential equation y\" + p(w)y' + My = 0, (:c > 0) (37) with 3:1)(33) 2 pa + p13: + and w2q(:r:) 2 go + qlm + having radii of convergence R1, R2 respectively. Let r1, m be the roots of the indicial equation 7'2 + (p3 - 1)?' + go 2 0, (33) where an 2 T2 if the roots are real. (Otherwise they are complex conjugates.) Seeking y(:c) in the form y(a:) = 1:" Z ants" :- Z anwn'\