Solution
= Solution
Integrating the equation through $t=1$ gives the <jump condition for an impulse>
$$
T_1(1^+)-T_1(1^-)=-1.
$$
For $1<t<2$ the equation is again homogeneous, so propagation of the post-jump temperature gives
$$
T_1(t)-T_\infty=(\Delta e^{-a}-1)e^{-a(t-1)}
=(\Delta-e^a)e^{-at}.
$$
Therefore
$$
\boxed{T_1(t)=T_\infty+(T_0-T_\infty-e^a)e^{-at}}.
$$