SYSTEM
I have a system of differential equations:
\begin{align} \frac{1}{2} (\sigma^x)^2 \frac{\partial^2 V_- (x_t)}{\partial x_t^2} + \left( \frac{\delta a_0}{a_1} - a_1 x_t \right) \frac{\partial V_- (x_t)}{\partial x_t} - e_- V_- (x_t) &= -\frac{+ 2x_t - 2 \lambda e_+ (a_0 + a_1 x_t)}{e_- - e_+} \\ \frac{1}{2} (\sigma^x)^2 \frac{\partial^2 V_+ (x_t)}{\partial x_t^2} + \left( \frac{\delta a_0}{a_1} - a_1 x_t \right) \frac{\partial V_+ (x_t)}{\partial x_t} - e_+ V_+ (x_t) &= -\frac{- 2x_t + 2 \lambda e_- (a_0 + a_1 x_t)}{e_- - e_+} \end{align}
GENERAL SOLUTION HOMOGENEOUS
To solve it, I start by finding the general solution of the homogeneous equations (I condense the two solutions into one by using the symbol $\pm$): \begin{align} V_\mp^g(x_t) = c_\mp^1 H_{-\frac{e_\mp}{a_1}}\left(\frac{a_1^2 x_t-a_0 \delta }{a_1^{3/2} \sigma^x }\right) + c_\mp^2 \, _1F_1\left(\frac{e_\mp}{2 a_1};\frac{1}{2};\frac{\left(a_1^2 x_t -a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2}\right) \end{align} where $H_n(x_t)$ is the Hermite function (i.e. the generalization of the $n$-th physicist's Hermite polynomial): \begin{align} H_n(x_t) = 2^n \sqrt{\pi} \left( \frac{1}{\Gamma \left( \frac{1-n}{2} \right)} \, _1F_1 \left( - \frac{n}{2}; \frac{1}{2}; x_t^2 \right) - \frac{2x_t}{\Gamma \left( -\frac{n}{2} \right)} \, _1F_1 \left( \frac{1-n}{2}; \frac{3}{2}; x_t^2 \right) \right) \end{align} and $_1F_1(a,b,x_t)$ is the Kummer confluent hypergeometric function: \begin{align} _1F_1(a,b,x_t) &= \sum_{k=0}^\infty \frac{\frac{\Gamma(a+k)}{\Gamma(a)} x_t^k}{\frac{\Gamma(b+k)}{\Gamma(b)} k!} \end{align}
PARTICULAR SOLUTION INHOMOGENEOUS
Then I find a particular solution of the inhomogeneous equation with the variation of parameters method: \begin{align} V_\mp^p(x_t) = - v_\mp^{g1}(x_t) \int \frac{v_\mp^{g2}(x_t) f_\mp(x_t)}{a_\mp(x_t) w_\mp(x_t)} dx + v_\mp^{g2}(x_t) \int \frac{v_\mp^{g1}(x_t) f_\mp(x_t)}{a_\mp(x_t) w_\mp(x_t)} dx \end{align} where $w_\mp(x_t)$ is the Wronskian: \begin{align} w_\mp(x_t) = v_\mp^{g1}(x_t)\frac{\partial v_\mp^{g2}(x_t)}{\partial x_t} - v_\mp^{g2}(x_t) \frac{\partial v_\mp^{g1}(x_t)}{\partial x_t} \end{align} and where: \begin{align} v_\pm^{g1} = H_{-\frac{e_\mp}{a_1}}\left(\frac{a_1^2 x_t-a_0 \delta }{a_1^{3/2} \sigma^x }\right) \end{align} \begin{align} v_\pm^{g2} = _1F_1\left(\frac{e_\mp}{2 a_1};\frac{1}{2};\frac{\left(a_1^2 x_t -a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2}\right) \end{align}
GENERAL SOLUTION INHOMOGENEOUS
Third, the general solution to the nonhomogeneous equation is equal to the sum of the general solution of the homogeneous equation and the particular solution of the nonhomogeneous equation: \begin{align} V_\mp(x_t) &= c_\mp^1 H_{-\frac{e_\mp}{a_1}}\left(\frac{a_1^2 x-a_0 \delta }{a_1^{3/2} \sigma^x}\right)+c_\mp^2 \, _1F_1\left(\frac{e_\mp}{2 a_1};\frac{1}{2};\frac{\left(a_1^2 x-a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2 }\right) \mp \\ & \mp H_{-\frac{e_\mp}{a_1}}\left(\frac{a_1^2 x_t-a_0 \delta }{a_1^{3/2} \sigma^x}\right) \frac{2 a_1^2}{e_\mp (e_--e_+)} \times \nonumber \\ & \times \int \frac{ (e_\pm \lambda (a_0+a_1 x_t)-x_t) \, _1F_1\left(\frac{e_\mp}{2 a_1};\frac{1}{2};\frac{\left(a_1^2 x_t-a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2 }\right) }{\left(a_1^2 x_t - a_0 \delta\right) H_{-\frac{e_\mp}{a_1}}\left(\frac{a_1^2 x_t-a_0 \delta }{a_1^{3/2} \sigma^x}\right) \, _1F_1\left(\frac{e_\mp}{2 a_1}+1;\frac{3}{2};\frac{\left(a_1^2 x_t-a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2 }\right) +a_1^{3/2} \sigma^x H_{-\frac{a_1+e_\mp}{a_1}}\left(\frac{a_1^2 x_t-a_0 \delta }{a_1^{3/2} \sigma^x}\right) \, _1F_1\left(\frac{e_\mp}{2 a_1};\frac{1}{2};\frac{\left(a_1^2 x_t-a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2 }\right)} \, dx_t \pm \nonumber \\ &\pm \, _1F_1\left(\frac{e_\mp}{2 a_1};\frac{1}{2};\frac{\left(a_1^2 x_t-a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2 }\right) \frac{2 a_1^2}{e_\mp (e_--e_+)} \times \nonumber \\ & \times \int \frac{ (e_\pm \lambda (a_0+a_1 x_t)-x_t) H_{-\frac{e_\mp}{a_1}}\left(\frac{a_1^2 x_t-a_0 \delta }{a_1^{3/2} \sigma^x}\right)}{\left(a_1^2 x_t - a_0 \delta\right) H_{-\frac{e_\mp}{a_1}}\left(\frac{a_1^2 x_t-a_0 \delta }{a_1^{3/2} \sigma^x}\right) \, _1F_1\left(\frac{e_\mp}{2 a_1}+1;\frac{3}{2};\frac{\left(a_1^2 x_t-a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2 }\right) + a_1^{3/2} \sigma^x H_{-\frac{a_1+e_\mp}{a_1}}\left(\frac{a_1^2 x_t-a_0 \delta }{a_1^{3/2} \sigma^x}\right) \, _1F_1\left(\frac{e_\mp}{2 a_1};\frac{1}{2};\frac{\left(a_1^2 x_t-a_0 \delta \right)^2}{a_1^3 (\sigma^x)^2 }\right)} \, dx_t \nonumber \end{align} It follows that: \begin{align} \textbf{J}(x_t) &= \textbf{Q} \textbf{V}(x_t) \\ \begin{bmatrix} J_{x_t}(x_t) \\ J_{\pi_t}(x_t) \end{bmatrix} &= \begin{bmatrix} e_- V_-(x_t) + e_+ V_+(x_t) \\ V_-(x_t) + V_+(x_t) \\ \end{bmatrix} \nonumber \\ \begin{bmatrix} J_{x_t}(x_t) \\ J_{\pi_t}(x_t) \end{bmatrix} &= \begin{bmatrix} e_- (V_-^g(x_t) + V_-^p(x_t)) + e_+ (V_+^g(x_t) + V_+^p(x_t)) \\ V_-^g(x_t) + V_-^p(x_t) + V_+^g(x_t) + V_+^p(x_t) \\ \end{bmatrix} \nonumber \\ \begin{bmatrix} J_{x_t}(x_t) \\ J_{\pi_t}(x_t) \end{bmatrix} &= \begin{bmatrix} e_- (c_-^1 v_-^{g1}(x_t) + c_-^2 v_-^{g2}(x_t) + V_-^p(x_t) ) + e_+ ( c_+^1 v_+^{g1}(x_t) + c_+^2 v_+^{g2}(x_t) + V_+^p(x_t) )\\ c_-^1 v_-^{g1}(x_t) + c_-^2 v_-^{g2}(x_t) + V_-^p(x_t) + c_+^1 v_+^{g1}(x_t) + c_+^2 v_+^{g2}(x_t) + V_+^p(x_t) \nonumber \end{bmatrix} \end{align} where: \begin{align} e_\pm = \frac{\delta \theta \pm \sqrt{\delta^2 \theta^2 + 4 \theta (\epsilon - 1) (1 + \varphi)}}{2 \theta} \nonumber \end{align} \begin{align} a_1 &= \frac{- \delta \theta + \sqrt{\delta^2 \theta^2 + 4 \theta (\epsilon - 1) (1 + \varphi)}}{2 \theta} \end{align} \begin{align} V = Q^{-1} J = \begin{bmatrix} V_- \\ V_+ \end{bmatrix} \end{align} \begin{align} Q^{-1} = \begin{bmatrix} e_- &e_+ \\ 1 &1 \end{bmatrix} \end{align} where $J$ is the value function of an optimization problem.
BOUNDARY CONDITIONS
Next, I impose boundary conditions to find the four constants of $V_\mp(x_t)$ and hence the full solution of $J_{x_t}(x_t)$ and $J_{\pi_t}(x_t)$. \begin{align} \lim_{x_t \to \hat{x}^+} J_{x_t}(x_t) &= 0 \\ e_- (c_-^1 v_-^{g1}(\hat{x}) + c_-^2 v_-^{g2}(\hat{x}) + V_-^p(\hat{x}) ) + e_+ ( c_+^1 v_+^{g1}(\hat{x}) + c_+^2 v_+^{g2}(\hat{x}) + V_+^p(\hat{x}) ) &= 0 \nonumber \end{align} \begin{align} \lim_{x_t \to \hat{x}^+} J_{x_t x_t}(x_t) &= 0 \\ e_- \left(c_-^1 \frac{\partial v_-^{g1}(\hat{x})}{\partial x_t} + c_-^2 \frac{\partial v_-^{g2}(\hat{x})}{\partial x_t} + \frac{\partial V_-^p(\hat{x})}{\partial x_t} \right) + e_+ \left( c_+^1 \frac{\partial v_+^{g1}(\hat{x})}{\partial x_t} + c_+^2 \frac{\partial v_+^{g2}(\hat{x})}{\partial x_t} + {\partial V_+^p(\hat{x})}{\partial x_t} \right) &= 0 \nonumber \end{align} \begin{align} \lim_{x_t \to \hat{x}^+} J_{\pi_t}(x_t) &= \frac{2 \theta}{(\epsilon - 1) (1 + \varphi)} x_t \\ c_-^1 v_-^{g1}(\hat{x}) + c_-^2 v_-^{g2}(\hat{x}) + V_-^p(\hat{x}) + c_+^1 v_+^{g1}(\hat{x}) + c_+^2 v_+^{g2}(\hat{x}) + V_+^p(\hat{x}) &= \frac{2 \theta}{(\epsilon - 1) (1 + \varphi)} \hat{x} \nonumber \end{align} \begin{align} \lim_{x_t \to \hat{x}^+} J_{\pi_t x_t}(x_t) &= \frac{2 \theta}{(\epsilon - 1) (1 + \varphi)} \\ c_-^1 \frac{\partial v_-^{g1}(\hat{x})}{\partial x_t} + c_-^2 \frac{\partial v_-^{g2}(\hat{x})}{\partial x_t} + \frac{\partial V_-^p(\hat{x})}{\partial x_t} + c_+^1 \frac{\partial v_+^{g1}(\hat{x})}{\partial x_t} + c_+^2 \frac{\partial v_+^{g2}(\hat{x})}{\partial x_t} + \frac{\partial V_+^p(\hat{x})}{\partial x_t} &= \frac{2 \theta}{(\epsilon - 1) (1 + \varphi)} \nonumber \end{align}
QUESTION
Notice that $-\frac{e_-}{a_1}$ is an integer but $-\frac{e_+}{a_1}$ is not. What does this imply about the validity of the solutions? Does this mean I can exclude the first term of the general solution for $V_+^g(x_t)$ because the degree of the Hermite polynomial is not an integer? I am confused because for a non-integer $n$ we can work with the Hermite function rather than the Hermite polynomial but I am not sure whether we can exclude the solution. Moreover, I does not seem to me that the boundary conditions can be satisfied only for special values of the index $n$. The reason I ask this question, is that if I were to keep the solutions as they are, I would be using all four available boundary conditions to determine the constants, and I would have no equation left to back out $\hat{x}$.