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.