1. The problem statement, all variables and given/known data I Have a differential equation y'' -xy'-y=0 and I must solve it by means of a power series and find the general term. I actually solved the most of it but I have problem to decide it in term of a ∑ notation! 2. Relevant equations y'' -xy'-y=0 3. The attempt at a solution I know the recurrence relation is an+2= an / (n+2) ´for the first solution y1 I choose even numbers for n and for the second solution y2 I choose odd numbers Even numbers: n=0 → a2= a0 / (0+2) , n=2 → a4= a0 / (2*2*2) n=4 → a6= a0 / (6*2*2*2) Odd numbers: n=1 → a3= a1 / (3) , n=3 → a5= a1 / (5*3) n=5 → a7= a1 / (7*5*3) For even numbers I see that it's not so hard to find : y1 = ∑(x2n/2n*n!) But I have problem with y2, I tried many times to find a solution but I faild! I see just in the answer y2 is : ∑(2n*n!*x2n+1/(2n+1)!) As I said I can solve this diff equation but I can't find the general solution for y2 in term of a ∑, What kind of algorithm must I use to get ∑(2n*n!*x2n+1/(2n+1)!) ?! Is ∑(2n*n!*x2n+1/(2n+1)!) the only solution for y2 or can we find a similar series which reaches the same result?