= Solution
The term $b_1E$ is the effective atmospheric addition from ongoing emissions. Its coefficient $0.25\,\mathrm{ppmv/GtC}$ is about half the full-retention estimate $0.47\,\mathrm{ppmv/GtC}$, so the model implicitly removes or partitions some emitted carbon promptly. The term $-b_0(C-C_0)$ represents slower net uptake of the excess by other reservoirs; its <relaxation time> is $b_0^{-1}=200$ years. Both terms have units of ppmv per year.
Thousands of years at constant $E_0$ erase the initial transient, giving
$$
0=b_1E_0-b_0(C-C_0),\qquad
\boxed{C(1800)=C_0+\frac{b_1E_0}{b_0}=279+1=280\,\mathrm{ppmv}.}
$$
The reference $C_0$ is the zero-anthropogenic-emission equilibrium, not the equilibrium when the stated small persistent emissions are present.
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