= Solution
Use <separation of variables>, $u=X(x)Y(y)$, and divide by $XY$ where nonzero. A real separation constant $\kappa$ gives
$$
\frac{X''}{X}=\frac{Y'}Y+1+\sin y=\kappa,
$$
so $X''=\kappa X$ and $Y'=(\kappa-1-\sin y)Y$. Normalizing the second factor to one at zero gives
$$
\boxed{Y_\kappa(y)=\exp[(\kappa-1)y+\cos y-1].}
$$
The spatial <ordinary differential equation> has three types of real solution:
$$
\boxed{X(x)=\begin{cases}A\cos(kx)+B\sin(kx),&\kappa=-k^2<0,\\A+Bx,&\kappa=0,\\A\cosh(kx)+B\sinh(kx),&\kappa=k^2>0.\end{cases}}
$$
Products $X Y_\kappa$ are the separated <eigenfunctions>. The negative, zero and positive separation constants give trigonometric, linear and <hyperbolic function> forms respectively. There is no discrete quantization until <boundary conditions> on $x$ are supplied. Since $\cos y-1$ is bounded, the envelope of $Y_\kappa$ decays for $\kappa<1$, grows for $\kappa>1$, and is bounded and periodic for $\kappa=1$. The scalar prefactor can also be derived by <removing a variable scalar reaction rate by an integrating factor>.
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