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$2\wedge 3\wedge 4$ 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 $\sqrt{2}$?

It's Only a Minus Sign

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?