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# Differential Equation Matcher

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Age 16 to 18

Challenge Level

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.

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?