Struggling with an Ugly Differential Equation? Try This Technique!

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Homework Help Overview

The discussion revolves around a second-order linear differential equation of the form y''(x)-(x+ln x)y(x)=0. Participants are exploring potential techniques for solving this equation.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants are considering various methods, including numerical solutions and the application of theorems related to linear ordinary differential equations. There is a question regarding the identification of particular solutions based on the given equation.

Discussion Status

The discussion is ongoing, with participants offering hints and exploring different approaches. One participant has provided a hint related to the behavior of the function (x+ln x), suggesting a potential avenue for further investigation.

Contextual Notes

One participant mentions feeling rusty, indicating a possible lack of familiarity with the techniques required for this type of problem. There may be constraints related to the level of detail expected in the responses, as it is a homework help context.

kuahji
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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?
 
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Numerical solution?
 
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?
 
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.
 

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