Applications of coupled systems of $\;2\times 2\;$ linear differential equations I am providing maths help to some students studying just before University level in mathematics. 

I am writing some practice questions for them on solving coupled first order linear equations and I was wondering if there are any 'real-life' applications of this. 

I am having difficulty finding examples because the systems they need to solve are restricted to two dependent variables and one independent variable and need to be linear. (Most of the easy go-to options, e.g. SIR models, are not linear). 

I have already used:


predator-prey problems
two-tanks mixing problems


Are there any others? 

[For information, the equations they are required to solve should be of the form 

$\frac{dx}{dt} = ax + by + f(t), \hspace{3mm} \frac{dy}{dt} = cx + dy + g(t)$.]
Coupled harmonic oscillators are a good example. The outer rectangles are walls, the inner ones are masses and the lines are springs.  The heavy lines are strong springs.  If you write it in terms of the positions of each mass, the coupling spring shifts the frequencies up and down a bit.  You can do the linear algebra to show that the modes are symmetric and antisymmetric and that the correct basis is in those terms.
Coupled harmonic oscillators are a good example. The outer rectangles are walls, the inner ones are masses and the lines are springs.  The heavy lines are strong springs.  If you write it in terms of the positions of each mass, the coupling spring shifts the frequencies up and down a bit.  You can do the linear algebra to show that the modes are symmetric and antisymmetric and that the correct basis is in those terms.