Let $M$ be an arbitrary real $2\times2$ matrix.

Show that if $x(t)$ $\land$ $y(t)$ $\in$ $C^1(\mathbb{R})$ and that if $a(t)$ $\land$ $b(t)$ $\in$ $C^0(\mathbb{R})$, then the following system of differential equations is linear:

$$\begin{bmatrix}
x\,'(t)\\
y\,'(t)\\
\end{bmatrix}=\mathbf{M}\begin{bmatrix}
x(t)\\
y(t)\\
\end{bmatrix}+\begin{bmatrix}
a(t)\\
b(t)\\
\end{bmatrix}$$

Hint: Show that the vector space transformation $f: C^1(\mathbb{R})\times C^1(\mathbb{R}) \rightarrow C^0(\mathbb{R})\times C^0(\mathbb{R})$ is given by $f(x) = x' - Ax$

@Astyx How about this one then:
In a big auditorium, a massive metal ball is placed in the middle of the room hanging from a thin wool thread from the ceiling. The temperature in the surrounding room is assumed to be the same throughout the room and is a function $R(t)$. At the time $t=0$, the room temperature is $R(0)=R_0$.

At the time $t = 0$, the temperature of a metal ball is $M(0)=T_0$

The two differential equations are:

$1) d/dt(M(t))=-k(M(t)-R(t)), t\geq 0, M(0)=T_0$ (k is the heat capacity)