Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# It's Only a Minus Sign

In this problem we shall see how a simple minus sign in a differential equation can completely change the character of the solution.

Two particles are released from $x = 1$ at time $0$ and their speed at any point x will be given by these two differential equations:

Now solve the equations to see if you were correct.

Would releasing the particles with a negative velocity from the origin have a significant effect on the type of motion which results?

Could you find initial starting points and velocities which would give rise to motions in which the particles slow down and stop?

## You may also like

### Circles Ad Infinitum

### Climbing Powers

Or search by topic

Age 16 to 18

Challenge Level

In this problem we shall see how a simple minus sign in a differential equation can completely change the character of the solution.

Two particles are released from $x = 1$ at time $0$ and their speed at any point x will be given by these two differential equations:

particle $A \quad \quad$ | $\frac{dx}{dt}=x$ |

particle $B$ | $\frac{dx}{dt}=-x$ |

Without solving the equations, can you describe how the
particles will move? Draw a sketch graph of the path you expect the
particles to take.

Now solve the equations to see if you were correct.

Next suppose that two more particles are released with a
positive velocity at time $0$ from the origin and move according to
these equations, in which v is the velocity of each particle:

particle $C \quad \quad$ | $\frac{dv}{dt}=x$ |

particle $D$ | $\frac{dv}{dt}=-x$ |

Without solving the equations, can you provide a clear
description of the subsequent motion of the particles?

Would releasing the particles with a negative velocity from the origin have a significant effect on the type of motion which results?

Could you find initial starting points and velocities which would give rise to motions in which the particles slow down and stop?

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 $\sqrt{2}$?