# Differential equation matcher

In this problem we describe $5$ physical processes and give $6$ differential equations (one is a rogue!)

There are 3 different parts to the problem

1) Can you match the equations to the processes. You will need to describe clearly in words why you think that the equation is the correct one.One of the equations is a rogue, which does not match .

2) How do you think the variable will change throughout time, based on your understanding of the process?

3) Solve the equations and plot the solutions to see if you were correct

Process A

A lump of radioactive material has a mass of $X(t)$. Its mass slowly decays over time.

Process B

There are some live bacteria $X(t)$ in a jar. Some bacteria are attacked by a medicine and die. Once dead, the medicine moves on to another target. During this process, the rest of the bacteria eat and replicate.

Process C

A ball is attached on opposite sides to pieces of elastic. The elastic is stretched out and one end fixed to the ground and the other end to the ceiling. The ball is pulled vertically down slightly and then released. Its displacement from the equilibrium is $X(t)$

Process D

A curling stone is slid along an ice rink. The distance travelled from the point of release is $X(t)$. Frictional forces cause the stone gradually to come to rest.

Process E

Water is pumped from a lake at a constant rate. The volume of water in the lake is $X(t)$

Equations, for positive constants $a$ and $b$:

equation $U\quad \quad$ | $\frac{d^2X}{dt^2}=-aX+b$ |

equation $V\quad \quad$ | $\frac{d^2X}{dt^2}=-aX$ |

equation $W\quad \quad$ | $\frac{dX}{dt}=-aX$ |

equation $X\quad \quad$ | $\frac{dX}{dt}=-b+aX$ |

equation $Y\quad \quad$ | $\frac{d^2X}{dt^2}=-a\frac{dX}{dt}$ |

equation $Z\quad \quad$ | $\frac{dX}{dt}=-a$ |

You can also download a Word document with the equations and processes ready for card sorting.

Think about the derivative as the rate of change. Would each part of the description have a positive (increasing) or negative (decreasing) effect on the quantity $X(t)$? How would the size of the change depend on the magnitude of $X(t)$ at any given moment?

Freddie from Almond Hill school noticed that a decreasing quantity requires a negative first derivative, which allowed him correctly to match the equation for radioactive decay. Can you use these sorts of ideas to match all of the equations?

Pete Pederson from the Acadia Summer AP Calculus AB Workshop was the first to correctly identify the matches. Can you see how he did it?

A - W

B - X

C - V

D - Y

E - Z

### 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 $X(t)$?