Strange behavior of solution to $- \frac{\partial^2 \psi\left(x\right)}{\partial x^2} - e^{-\alpha x} \psi =0$

Question

Say we have the following equation the following equation (inspired from quantum mechanics with $V\left(x\right) = - e^{- \alpha x}$):

\begin{align} \label{eq:first-eq} - \frac{\partial^2 \psi\left(x\right)}{\partial x^2} - e^{-\alpha x} \psi =0 \tag{1} \end{align}

According to Mathematica the solutions are linear combinations of Bessel functions:

\begin{align} \label{eq:second-eq} \psi\left(x\right) = A J_{0} \left(\frac{2}{\alpha }e^{-\alpha x/2} \right) + B Y_{0} \left(\frac{2}{\alpha }e^{-\alpha x/2} \right) \tag{2} \end{align}

The code:

eq = -D[ψ[x], {x, 2}] - E^(-α*x)*ψ[x] == 0
DSolve[eq, ψ[x], x]
{{ψ[x] -> 
BesselJ[0, (2 Sqrt[E^(-x α)])/α] C[1] + 2 BesselY[0, (2 Sqrt[E^(-x α)])/α] C[2]}}

However, some is strange about this solution. Specifically notice that if we consider the equation:

$$- \frac{\partial^2 \psi\left(x\right)}{\partial x^2} - e^{\alpha x} \psi =0 $$

Which is Eq. \ref{eq:first-eq} but with $\alpha \rightarrow -\alpha $, then we should get Eq. \ref{eq:second-eq} with the $\alpha \rightarrow -\alpha $. However, instead we get:

$$ \psi\left(x\right) = A J_{0} \left(\frac{2}{\alpha }e^{\alpha x/2} \right) + B Y_{0} \left(\frac{2}{\alpha }e^{\alpha x/2} \right) $$

eq = -D[ψ[x], {x, 2}] - E^(α*x)*ψ[x] == 0
DSolve[eq, ψ[x], x]
{{ψ[x] -> 
{{ψ[x] -> BesselJ[0, (2 Sqrt[E^(x α)])/α] C[1] + 2 BesselY[0, (2 Sqrt[E^(x α)])/α] C[2]}}

I know that $J_{0}$ is even but $Y_{0}$ is not, what is happening? Also, which of the solutions is actually correct?

Answer 1

They are both correct:

eq = -D[ψ[x], {x, 2}] - E^(-α*x)*ψ[x] == 0;
DSolve[eq, ψ, x];
FullSimplify[eq /. First[%]]

(*Out[70]= True*)

eq = -D[ψ[x], {x, 2}] - E^(α*x)*ψ[x] == 0;
DSolve[eq, ψ, x];
FullSimplify[eq /. First[%]]

(*Out[67]= True*)

The parity of BesselJ and BesselY is completely irrelevant, given that their arguments are neither even or odd in α. And since you have two free constants any way, what's the contradiction?