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Use primes for differentiation with respect to $X$. Since $X=ϵx$, substituting the amplitude-phase representation into the differential equation and separating real and imaginary parts givesThe second equation is exactly $(\sigma A^2{)}^{\prime }=0$. Thus, with a real constant $C$ and $A\ne 0$,This exact amplitude-phase equation is the Ermakov-Pinney equation for the oscillator. Conversely, any real nonvanishing $A$ satisfying it and $\sigma =C/A^2$ gives an exact complex solution. Existence is not an asymptotic assumption: choose independent real solutions $u,v$ of the original equation, form $y=u+iv$, and set $A=(u^2+v^2{)}^{1/2}$. Their nonzero constant Wronskian prevents simultaneous zeros. The phase derivative satisfies $\sigma A^2=u{v}_x-u_{x}v$, so it is constant. Complex conjugation gives a second branch, and linear combinations recover real solutions.

For the WKB approximation, work on an interval where $K=|k|>0$ is smooth and bounded away from zero. Choose the positive phase branch and an $ϵ$-independent positive $C$. The equations naturally admit even-power expansionsThere is no order-$ϵ$ phase correction in this convention. Leading order gives ${\sigma }_0=K$ and $A_0=(C/K{)}^{1/2}$. Comparing order ${ϵ}^2$ in the real equation yieldsConsequentlyThe constants absorb the normalization. The opposite phase branch changes the sign of $\sigma$ and its correction. A zero of $k$ is a classical turning point where this particular expansion fails; the exact amplitude-phase equation itself remains valid for a suitable complex solution.

For the Sturm-Liouville problem, take $ϵ={\lambda }^{-1/2}$ and use the interval coordinate as $X$, so $K(X)=X+1$ on $[0,1]$. Combining the conjugate branches to satisfy the left Dirichlet boundary condition produces a sine of the phase integral from zero. The right Dirichlet boundary condition requires an integer multiple of $\pi$. No turning-point phase shift is present. We haveThus the two-term Dirichlet WKB quantization isSolving first for the square root and then squaring givesAs an independent sign check, the Liouville transformation of a second-order equation with $t=X+X^2/2$ and $v=\sqrt{X+1}y$ gives $-v_{tt}-3v/[4(2t+1{)}^2]=\lambda v$ on an interval of length $3/2$. The mean transformed potential is $-3/16$, consistent with the constant spectral correction.