Solving Initial Value Problem using Power Series Method

[waqasyaqoob]

Solve the following initial value problem using a power series representation of the solution around x=0. Find the recurrence relation and the first five nonzero terms of the series solution.

d^2y/dx^2 + (2+x) dy/dx +4y=0 ; y(0)=1 ,y'(0)=0

 

[Mandlebra]

If y is a power series what does y look like?

 

[HallsofIvy]

First this is clearly a "homework or classwork" type problem so I am moving it.
Second, you have shown no attempt yourself to do this.

I know several very different ways to do this, but, because you hav not shown any work yourself, I have no idea which of them is appropriate for you.