Stuck on this differential equation I want to solve this differential equation with the power-series method: $$ x^{2}\cdot y''(x) +(1+x^{2}) y'(x) + y(x) =0$$ where $$y(0)=1$$
They want the solution given in elementary functions. I managed to get the recursive formula for $$j\ge 2$$$$a_{j+1} = \frac{-((j(j-1)+1)a_{j} + (j-1)a_{j-1})}{j+1} $$ 
And also got $$a_{0}=y(0)=1 $$,and then$$  a_{1}=-1, a_{2}=1/2 $$
But Im not sure how to answer in an elementary function, I tried to get some more values and its just irrational values. Appreciate help.

I agree with you're recurrence relation $(i+1)a_{i+1}=-((i^2-i+1)a_i+(i-1)a_{i-1})$. & the first few values $a_0=1, a_1=-1, a_2=1/2, a_3=-1/6$ from this I guess the general formulae $a_i=(-1)^i/i!$. You can easily show (by induction) that this is indeed true. Now the solution is obviously $y=e^{-x}$ & this does indeed satisfy the original equation.    
