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I want to solve the following ordinary differential equation $$u''(x)+\frac{x}{x+h}u(x)=0,$$ where $h>0$ is a constant. I would like to know which method for solution is appropriate for the above equation and if the solution can be expressed by elementary functions? The best I could accomplish is a messy looking series representation of the solution. So I am looking for more clever ways to attack the equation.

Best wishes

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    This looks to be a very difficult equation to solve. I guess that special functions would be involved in the solution. – 2017-02-02
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    Thank you for your help. Your comment helps a lot. – 2017-02-02
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    Meanwhile, I checked : guess what ! It is a **monster** ! Cheers. – 2017-02-02

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Claude Leibovici is right. The solution cannot be expressed with a finite number of elementary functions.

The functions involved are of the confluent hypergeometric kind (see below).

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