Below is another version of the creme brulee model. $k_{11}, k_{12},k_{21},k_{22}$ and $A$ are all positive constants. Show the the solutions to these ODEs can never oscillate (proof for $T_1$ is sufficient). In other words, the temperature of the caramelized sugar can only behave as overdamped response.
\begin{equation*} T_{1}'=k_{11}(A-T_1)+k_{12}(T_2-T_1) \end{equation*} \begin{equation*} T_{2}'=k_{21}(A-T_2)+k_{22}(T_1-T_2) \end{equation*}
Rewrite the system as
\begin{eqnarray*} T_{1}'=k_{11}A-(k_{11}+k_{12})T_1+k_{12}T_2\\ T_{2}'=k_{21}A+k_{22}T_1-(k_{21}+k_{22})T_2 \end{eqnarray*} and again in a matrix form $$ \begin{bmatrix} T_1\\ T_2 \end{bmatrix}' = \begin{bmatrix} k_{11}+k_{12}&k_{12}\\ k_{21}+k_{22}&k_{22} \end{bmatrix} \begin{bmatrix} T_1\\ T_2 \end{bmatrix} + A \begin{bmatrix} k_{11}\\k_{21} \end{bmatrix} $$ The eigenvalue/eigen vectors of the matrix are $$\lambda_{\pm}=\frac{1}{2}(k_{11}+k_{12}+k_{22}\pm D)$$ and $$v_{\pm}=\begin{bmatrix} \frac{1}{2}\frac{k_{11}+k_{12}-k_{22}\pm D}{k_{21} + k_{22}} \\1\end{bmatrix}$$ where $D=\sqrt{(k_{11} - k_{22})^2 + 2 k_{11} k_{12} + k_{12}^2 + 4 k_{12} k_{21} + 2 k_{12} k_{22} }$ and the solution to the homogeneous system ($A=0$) is given by $$a v_+e^{\lambda_+t} + b v_-e^{\lambda_-t} $$ which can only oscillate if $\lambda$ is complex. Since all the coefficients are positive, this can be only if $D$ is complex, i.e. its argument is negative.
Since argument of $D$ is sum of positive numbers, $D$ is real - you done.
Note, since the inhomogeneous part is constant it won't make it oscillatory.