Introduction to systems of ODEs

Suppose two masses on carts on frictionless surface are at displacements $x_1$ and $x_2$ as in an example in the source text. Suppose initial displacement is $x_1(0)=x_2(0)=0$, and initial velocity is $x_1'(0) = x_2'(0) = a$ for some number $a$. Use your intuition to solve the system, explain your reasoning.

Suppose the tanks are as in an example in the source text except that clean water flows in at a rate of $s$ liters per second into tank 1, and brine flows out of tank 2 and into the sewer also at a rate of $s$ liters per second. The rate of flow from tank 1 into tank 2 is still $r$, but the rate of flow from tank 2 back into tank 1 is $r-s$ (assume $r > s$).

1. Draw the picture.
2. Set up the system of equations.
3. Intuiti…