= Solution
For a small additional carbon <mass>, logarithmic forcing has <derivative> inversely proportional to the background atmospheric carbon inventory. With other conversion factors fixed, $b_0$ therefore decreases as background concentration rises. Solve the <temperature> equation using an <integrating factor> and the pulse burden:
$$
T(t)=\frac{b_0}{2}\left[\frac{1-e^{-b_1t}}{b_1}+\frac{e^{-2kt}-e^{-b_1t}}{b_1-2k}\right].
$$
At $b_1=2k$ the second term is its continuous limit $te^{-b_1t}$. For $0<k\ll b_1$, <temperature> initially rises over $b_1^{-1}$ toward roughly $b_0/b_1$, then relaxes over $(2k)^{-1}$ toward half that value. This produces an overshoot, rather than permanent growth. The two numerical limits requested separately below also follow directly from the <differential equation>.
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