Series solution to DE about ordinary point

[lordsurya08]

Homework Statement

Find two power series solutions of the DE

(x+2)y'' + xy' - y = 0

about the ordinary point x = 0 . Include at least first four nonzero terms for each of the solutions.

2. The attempt at a solution

I distributed the y'' term and substituted
y = Ʃ₀^inf c_nxⁿ
and its derivatives into the DE. I equated it to 0 and got two equations:

2c₂ - c₀ = 0

xⁿ(c_n+1*n(n+1) + c_n+2*(n+1)(n+2)+ c_n*(n-1)) = 0

The weird thing is that the second equation (the recurrence relationship) has three c terms in it, although the examples shown have two. How do I get c₂ and c₀? After that happens should I simply solve for c1 using the recurrence relationship with n = 0?

 

[vela]

You seem to have dropped the factor of 2 from the 2y'' term, so your two equations are slightly wrong.

Once $c_0$ is set, you can determine $c_n$ for $n \ge 2$ through the recurrence relation. That's your first solution.

The coefficient $c_1$ is still arbitrary. That corresponds to your second solution.