I have the following system of two second order ODEs:
\begin{align} (L -D)V+Q^{-1}g(x)=0 \end{align}
where
\begin{align} L = \frac{1}{2} \sigma_{x}^{2} I \partial_{x}^{2}+(A - a_1 x) I \partial_{x} \end{align}
\begin{align} D=diag(a_{1,-},a_{1,+})\\ \end{align}
\begin{align} Q=\left( \begin{array}{cc} a_{1,-}& a_{1,+} \\ 1 & 1 \end{array} \right) \end{align}
\begin{align} J = (J_{x}(x),J_{\pi}(x))^\top \end{align}
\begin{equation} V=Q^{-1}J \end{equation}
\begin{align} g(x) = 2( \lambda_{x} x,\lambda_{\pi} (a_{0} + a_{1}x))^{\top} \end{align}
To solve it, I need to find the complementary function and the particular integral.
I know that the result for the complementary function is as follows: \begin{align} \boldsymbol{V}_{1}=f_{1,-}F_{1,-}(x)+f_{2,-}F_{2,-}(x),\\ \boldsymbol{V}_{2}=f_{1,+}F_{1,+}(x)+f_{2,+}F_{2,+}(x), \end{align} where \begin{align} F_{1,\pm}(x)&=H_{-\frac{a_{1,\pm}}{a_{1}}}\left(\frac{\sqrt{a_{1}} x}{\sigma_{x} }-\frac{A}{\sqrt{a_{1}} \sigma_{x}}\right)\\ F_{2,\pm}(x)&=\, _1F_1\left(\frac{a_{1,\pm}}{2 a_{1}};\frac{1}{2};\left(\frac{\sqrt{a_{1}} x}{\sigma_{x} }-\frac{A}{\sqrt{a_{1}} \sigma_{x} }\right)^2\right) \end{align} where $H_{n}(z)$ is the $n$'th degree Hermite polynomial and $_1F_1(a,b,z)$ is the Kummer confluent hypergeometric function.
But I am very confused about how to do this first step. I try to solve the homogeneous equation:
\begin{align} (L -D)V=0 \end{align}
But I struggle to relate the result to the Hermite polynomial and the Kummer confluent hypergeometric function. Is this first step the right one?