### Why do this problem

Thinking about building equations will give a real boost to
students' understanding of differential equations as dynamical
processes in which some quantity undergoes a continuous
change.

Students will need to appeal to basic physical principles and the
concepts of rates of change to match all of the equations.

### Possible Approach

This problem would work well as a card-sorting activity and a
starter.

In each equation there are approximations made, which should be
discussed as a group.

This could lead to stimulating discussion in which students become
aware that the equations studied at school are in fact simple
approximations to real world problems. The skill of the
mathematical physicist is in understanding when an equation gives
good answers and when an equation breaks down and needs
refinement.

### Key Questions

- What do the equations say in words?
- Do you expect the derivatives to be positive or negative for
positive and negative values of $x$? Why?
- Is the assumption that $X(t)$ is any number sensible? If not
would it still give reasonable solutions to the modelling
problem?
- Are there other effects in the evolution of $X$ which could be
added to the equation? Would these effects be small or large? How
could these effects be incorporated?
- Is it reasonable that the constant terms are indeed constant,
or is it more realistic that they might vary at some point in the
evolution of $X(t)$?

### Possible Extension

Can students invent their own pairs of process and sensible
matching equation?

### Possible Support

Try the introductory problem

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