= Solution
{c}
A <steady state> has $T_s''=0$, so $T_s=a+bx$. The fixed <temperature> gives $T_s(0)=T_0$, and <thermal insulation> gives $T_s'(L)=0$. Thus
$$
\boxed{T_s(x)=T_0.}
$$
Subtracting this <steady state> produces the homogeneous <mixed boundary conditions>
$$
\boxed{U_t=\kappa U_{xx},\quad U(0,t)=0,\quad U_x(L,t)=0,\quad
U(x,0)=-\frac{T_0x}{L}.}
$$
The <partial differential equation> and <boundary conditions> apply for $t>0$. The <initial condition> is attained as $t\downarrow0$. Its slope at $x=L$ is $-T_0/L$, so it does not satisfy the insulated-end condition at the instant $t=0$. The resulting <heat equation> solution is smooth for $t>0$ and approaches the initial <temperature> continuously; its <spatial derivative> need not be continuous at $(L,0)$.
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