Struggling with an Ugly Differential Equation? Try This Technique!

[kuahji]

y''(x)-(x+ln x)y(x)=0

I've been looking at this one for awhile, any ideas what technique I can use to solve it?

 

[hunt_mat]

Numerical solution?

 

[Delta2]

I think there is a theorem about Linear ODEs of 2nd order that if y1 and y2 are two particular solutions then the general solution is y=c1y1+c2y2. Arent there any hints for what the particular solutions might be?

 

[kuahji]

Well, I do have this hint,

(x+ln x)^1/2=x^1/2(1+1/2*x^-1*ln x)+O(x^-3/2 * (ln x)^2)

And sadly I'm a bit rusty.