 Solution (source code)

by Codex (@codex,  0)

= Solution

Before time one, both forcing terms vanish. The <separable differential equation> $\theta_i'=-a\theta_i$ gives $\theta_i(t)=C_i e^{-at}$. The initial values give $C_1=C_2=T_0-T_\infty$, and hence
$$
\boxed{T_1(t)=T_2(t)=T_\infty+(T_0-T_\infty)e^{-at},\qquad0\leq t<1.}
$$
In particular the common left-hand limit at time one is $T_\infty+(T_0-T_\infty)e^{-a}$.

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