= Solution
{c}
Write $u_h^m=\sum_iU_i^m\varphi_i$ and define the <mass matrix> and reaction-diffusion <stiffness matrix>
$$
M_{ij}=\int_0^1\varphi_i\varphi_j,\qquad
S_{ij}=\int_0^1(\varphi_i'\varphi_j'+e^x\varphi_i\varphi_j).
$$
Then each step is
$$
\boxed{AU^{m+1}=(M/\Delta t-S/2)U^m,\qquad
A=M/\Delta t+S/2.}
$$
Both component matrices are symmetric, so $A$ is symmetric. For a nonzero coefficient vector $v$, its <finite element> <function> $v_h$ is nonzero and
$$
v^TAv=\Delta t^{-1}\|v_h\|_2^2+\tfrac12a(v_h,v_h)>0.
$$
Thus $A$ is also positive definite and every update has a unique solution. Because nonneighboring hats have disjoint interior supports, the matrix is tridiagonal in nodal order.
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