Series solution near an ordinary point

[hbomb]

some help with series solutions

I'm needing help on series solutions. It's been a while since I worked on them.

Find
\phi''(x_{0})
\phi'''(x_{0})
\phi''''(x_{0})


y"+xy'+y=0; y(0)=1. y'(0)=0

 

[Pyrrhus]

is \phi (x) a solution of the initial value problem?

if it its then start with

\phi '' (x) + x \phi ' (x) +\phi (x) = 0

 

[HallsofIvy]

and
\phi'''(x)+ x\phi''(x)+ \phi'(x)= 0
\phi''''(x)+ x\phi'''(x)+ \phi''(x)= 0

 

[hbomb]

I have a question that involves using those derivatives. It's asks to determine the values at those derivatives with y(0)=1, y'(0)=1

When I used those numbers I didn't get the correct answer, which is -1, 0, 3. I think the derivatives are wrong.