= Solution
Put $\vartheta=T-T_0$ and assume $C>0$, $\lambda>0$. The <linear differential equation> for a constant forcing step is
$$
\dot\vartheta+\frac\lambda C\vartheta=\frac{F_0}{C},\qquad\vartheta(0)=0.
$$
Multiplying by the integrating factor $e^{\lambda t/C}$ and integrating from zero to $t$ gives
$$
\boxed{T(t)=T_0+\frac{F_0}{\lambda}(1-e^{-k_Tt}),\qquad k_T=\frac\lambda C.}
$$
The response initially grows at rate $F_0/C$ and tends to the equilibrium warming $F_0/\lambda$. Heat capacity per unit area and radiative-feedback units make $k_T$ an inverse time.
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