Conditions of a differential equation Consider the differential equation
\begin{align}
2 x^2 y'' + x(x^2 - 1) y' + (2 x^2 - x +1)y = 0 \hspace{5mm} y(0) = 0, y'(0)=1.
\end{align}
A solution readily found is
\begin{align}
y(x) &= B_{0} \, \sqrt{x} (x+1) \, e^{- \frac{x^{2}}{4}} \\
& \hspace{5mm} + B_{1} \, e^{- \frac{3 \pi i}{4}} \, \sqrt{x} \, e^{- \frac{x^{2}}{4}} \, \left[ 4 \, e^{\frac{3 \pi i}{4}} \, x \, e^{- \frac{x^{2}}{4}}
+ \sqrt{2} (x+1) \left( 2 \Gamma\left(\frac{3}{4}, - \frac{x^{2}}{4}\right) - i \, \Gamma\left(\frac{1}{4}, - \frac{x^{2}}{4}\right) \right) \right]
\end{align}
How are the boundary conditions to be aplied such that $B_{0}$ and $B_{1}$ can be obtained?
 A: We develop $y(x)$ in the vicinity of $x=0$
$$2\Gamma\left(\frac{3}{4},-\frac{x^2}{4}\right)+i\Gamma\left(\frac{1}{4},-\frac{x^2}{4}\right)=2\Gamma\left(\frac{3}{4}\right)+i\Gamma\left(\frac{1}{4}\right)-(2i)^{3/2}\sqrt{x}+O(x)$$
$$4e^{\frac{3\pi i}{4}}e^{-\frac{x^2}{4}}\sqrt{2}(x+1)\left(2\Gamma\left(\frac{3}{4},-\frac{x^2}{4}\right)+i\Gamma\left(\frac{1}{4},-\frac{x^2}{4}\right)\right)=$$ $$=2\Gamma\left(\frac{3}{4}\right)+i\Gamma\left(\frac{1}{4}\right)-(2i)^{3/2}\sqrt{x}+O(x)$$
$$e^{-\frac{3\pi i}{4}}\sqrt{x}e^{-\frac{x^2}{4}} \left[4e^{\frac{3\pi i}{4}}e^{-\frac{x^2}{4}}\sqrt{2}(x+1)\left(2\Gamma\left(\frac{3}{4},-\frac{x^2}{4}\right)+i\Gamma\left(\frac{1}{4},-\frac{x^2}{4}\right)\right)\right]=$$ $$= e^{-\frac{3\pi i}{4}}\left(2\Gamma\left(\frac{3}{4}\right)+i\Gamma\left(\frac{1}{4}\right)\right)\sqrt{x} -2^{3/2}x +O(x^{3/2})$$
$$\sqrt{x}(x+1)e^{-\frac{x^2}{4}}=\sqrt{x}+O(x^{3/2})$$
$$y(x)=B_0\sqrt{x}+B_1\left(e^{-\frac{3\pi i}{4}}\left(2\Gamma\left(\frac{3}{4}\right)+i\Gamma\left(\frac{1}{4}\right)\right)\sqrt{x}-2^{3/2}x\right) +O(x^{3/2}) $$
The condition $y(0)=0$ is fulfilled.
In order to have a finite value of $y'(0)$ it is necessay that there is no term $\sqrt{x}$ because, if not the derivative $\frac{1}{2\sqrt{x}}$ would be infinite. Hense :
$$B_0+B_1e^{-\frac{3\pi i}{4}}\left(2\Gamma\left(\frac{3}{4}\right)+i\Gamma\left(\frac{1}{4}\right)\right) =0$$
$$y(x)=-B_1 2^{3/2}x +O(x^{3/2}) $$
and the condition $y'(0)=1$ leads to :
$$B_1=-2^{-3/2}$$
$$B_0=2^{-3/2}e^{-\frac{3\pi i}{4}}\left(2\Gamma\left(\frac{3}{4}\right)+i\Gamma\left(\frac{1}{4}\right)\right)$$
