 Solution

Choose $\delta =\sqrt{ϵ}$ in the WKB approximation and substitute $y=\exp (S_0/\delta +S_1+\cdots )$. The leading equation isso take $S_0=\pm 2x^{3/2}/3$. At the next order,givingup to an absorbed constant. Label the recessive and dominant branches respectively byThe formal outer solution is $c_1{y}_1+c_2{y}_2$. The point zero is a simple wave turning point: $S_0^{\prime }$ vanishes and the WKB amplitude diverges.

The boundary condition at infinity selects $c_2=0$, with $c_1$ to be fixed by matching. This statement concerns the actual recessive branch. One cannot literally send $x\to \infty$ in the truncated small-$ϵ$ formulas: their ordering fails when $ϵx$ is no longer small, and their common Gaussian factor would then spuriously suggest that both branches decay.

To justify the selection, remove the first derivative exactly with $y=e^{-x^2/4+x/2}v$. ThenThe coefficient is positive for $x\ge 0$, so it has distinct dominant and recessive branches. For very large $x$, the dominant branch restores $y$ growth of exponential rate $1/ϵ$, while the recessive branch decays. Matching to this recessive branch removes $c_2$.