Linear systems of differential equations

Question

I would like to see an example of a real physical situation where one can find a set of variables evolving according to a system of linear differential equations.

I wasn't able to find any such example so I'm asking here.

Answer 1

When you have a point mass $m$ that can move in the (horizontal) $y$-direction, but is drawn to the origin by a spring then it moves according to the ODE $$m\>\ddot y=-f\> y\ ,$$ where $f>0$ denotes the spring constant. Assume now that you have two such masses with origins at a certain distance of each other. Then they independently move according to the system of ODEs $$m_1\>\ddot y_1=-f_1\> y_1\ ,\qquad m_2\>\ddot y_2=-f_2\> y_2\ .\tag{1}$$ To make things more interesting we now connect the two masses by a very weak "coupling" spring, whereby we assume that the system is in equilibrium when $y_1=y_2=0$. The ODE system $(1)$ then has to be replaced by $$m_1\>\ddot y_1=-f_1\> y_1+\kappa(y_2-y_1)\ ,\qquad m_2\>\ddot y_2=-f_2\> y_2-\kappa(y_2-y_1)\ ,\tag{2}$$ where $\kappa>0$ encodes the constant of the "coupling" spring. Introducing velocities one then can convert $(2)$ into the system $$\left.\eqalign{\dot y_1&=v_1 \cr \dot v_1&=-(f_1+\kappa)y_1+\kappa y_2 \cr \dot y_2&= v_2 \cr \dot v_2&=\kappa y_1-(f_2+\kappa)y_2\cr}\right\}\ .$$ The latter is a system of the form envisaged in the question.