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How to solve the following ordinary differential equation \begin{equation} f_{xx}^{^{\prime \prime }}+2axf_{x}^{^{\prime }}-2sf=0 \end{equation} with $a$ and $s$ are arbitrary real numbers and the boundary conditions \begin{equation} f\left( \delta \right) =f\left( -\delta \right) =1 \end{equation}

After checking some books, I got the general solution \begin{equation} f\left( x\right) =C_{1}\Phi \left( -\frac{s}{2a},\frac{1}{2},-ax^{2}\right) +C_{2}\Psi \left( -\frac{s}{2a},\frac{1}{2},-ax^{2}\right) \end{equation} where $\Phi$ and $\Psi$ are degenerate hypergeometric functions. Are there any other forms of the solution, such as double integral or serious?

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    I'm familiar with $f_x$ as notation for the derivative of $f$, likewise $f'$, but $f_x'$ is a new one.2012-12-11
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    It means the derivative of $f$ with respect to $x$. It is used in some books, such as "Handbook of Exact Solutions for Ordinary Differential Equations" by Andrei D. Polyanin and Valentin F. Zaitsev.2012-12-11
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    Of course there are other forms of the solution, you should find those by another methods, see http://math.stackexchange.com/questions/97586 for details. Even $\Phi\left(-\dfrac{s}{2a},\dfrac{1}{2},-ax^2\right)$ and $\Psi\left(-\dfrac{s}{2a},\dfrac{1}{2},-ax^2\right)$ themselves are in fact have other forms, as they are only symbols.2012-12-11
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    Don't limit yourself on the view on only known special functions, http://tw.knowledge.yahoo.com/question/question?qid=1011072501482 is a very good example that the ODE whose the solution have quite nice forms of non-known special functions rather than known special functions.2012-12-11
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    @doraemonpaul: I checked your method math.stackexchange.com/questions/97586. It may not apply here. Since in this problem, the parameter $a$ may be zero.2012-12-11
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    @Xiangyu Meng: The case of $a=0$ should be separated from the cases of $a\neq0$ as $a=0$ will make the form of the ODE having large difference. Moreover, even $\Phi\left(-\frac{s}{2a},\frac{1}{2},-ax^{2}\right)$ and $\Psi\left(-\frac{s}{2a},\frac{1}{2},-ax^{2}\right)$ will have problems when $a=0$ .2012-12-12

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