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I try to solve this differential equation whose solution seems not to be constructable in power series $y''+(x+a/x^2+b)y=0$, where $a$ and $b$ are some positive real numbers. If one can help me please?

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Is the coefficient of $y$ supposed to be $x+\frac{a}{x^2} + b$, or $\frac{x+a}{x^2+b}$? The slash notation is ambiguous; please avoid it. –  Arturo Magidin Feb 3 '12 at 20:27
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The differential equation $$y''+ \left(x+ \frac{a}{x^2}+b \right)y=0$$ has a regular singular point at $x=0$. In such a case, it is not always possible to construct a power series solution. However, it is always possible to find a solution of the form $$y = x^\alpha p(x)$$ with $\alpha \in \mathbb{C}$ and $p(x) = \sum_{n=0} p_n x^n$.

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And what if OP really meant $$y''+{x+a\over x^2+b}y=0$$ –  Gerry Myerson Mar 4 '12 at 23:51
    
@Gerry: then he should be able to develop $y(x)$ in a power series around $x=0$. –  Fabian Mar 5 '12 at 3:11
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