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Write $E=\mathrm{erf}z$, $g=e^{-z^2}$ and $b=2/\sqrt{\pi }$. The first correction in the inner expansion satisfies ${\mathcal{L}}_0{Y}_1=2zE$. Direct differentiation verifies that $zE+bg$ is a particular solution. Adding the constant and error function homogeneous solutions gives $Y_1=zE+bg+A+BE$.

The left boundary condition forces $A=-b$. Re-expanding the outer expansion at $x=sz$ givesTherefore $Y_1\sim z$, with no constant term in its large-$z$ overlap. Since $E\to 1$ and $g\to 0$, this sets $B=b$. Thus the required half-order correction isIt vanishes at zero and has the correct linear overlap. Keeping this $O(\sqrt{\epsilon })$ term is essential even though the outer solution has no correction of that order.

For either scalar equation, the complementary solution isso initial velocity transients decay when $t\gg (ϵ\Omega {)}^{-1}$. In the exact corrected forced dust epicycle with aerodynamic drag, $v_{x,p}^{\prime }=u\cos \theta$ is a particular solution, and the original radial equation then gives $v_{y,p}^{\prime }=u\sin \theta /2$. ConsequentlyThis is resonant dust entrainment by an epicyclic gas wave: in the adopted long-wavelength gas approximation, the forcing is a free epicycle of the dust too, so matching the gas makes the drag zero. The convergence holds for every $ϵ>0$ in this linear model; at $ϵ=0$ transients never decay, so the zero-drag and late-time limits do not commute.

If instead the printed scalar equation is treated literally for finite $ϵ$, its particular solution isIt has amplitude $|u|/\sqrt{1+{ϵ}^2/4}$ and phase offset $\arctan (ϵ/2)$, not exact phase/amplitude agreement. Agreement in that truncated equation means only leading order as $ϵ\to 0$. This distinguishes the intended approximation from a false exact reading of the printed assertion.

There is a further order-of-limits condition: a real finite-wavelength gas wave has $\omega -\Omega \ne 0$. Leading resonant entrainment over the damping time requires $|\omega -\Omega |\ll ϵ\Omega$, or $(k{c}_s/\Omega {)}^2\ll ϵ$. Otherwise solve the dust transfer equations at the actual $\omega$; detuning can reduce the dust response even though $|k|c_s/\Omega \ll 1$.

The identityshows that $y_n^{(p)}=h^3{n}^3/6$ is a particular solution. Combining it with the homogeneous solution from part (b) givesThe condition $y_0=0$ gives $A=0$, and $y_{-1}=y_1$ gives $B=-h^3/6$. HenceThe exact initial-value solution is $y(t)=t^3/6$. Since $N=1/h$,Thus the absolute endpoint error is $h^2/6$, consistent with the second-order truncation error of the centred scheme.

The homogeneous solution isFor the constant and $\cos 2\varphi$ forcing terms, a particular solution isIndeed, applying $d^2/d{\varphi }^2+1$ multiplies the $\cos 2\varphi$ coefficient by $-3$. ThusThe $B\sin \varphi$ term only changes the definition of $b$ at this order, so one may set $B=0$.

For $y\prime -ay=e^{\lambda x}$, subtracting an appropriate homogeneous solution from the particular solution removes its apparent singularity as $\lambda \to a$. The resulting derivative in the parameter produces the resonant particular solution $x{e}^{ax}$. The arbitrary constant must be reparametrized before taking the limit.