It's only a minus sign
Solve these differential equations to see how a minus sign can change the answer
Problem
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 $B$ | $\frac{dx}{dt}=-x$ |
Now solve the equations to see if you were correct.
| particle $C \quad \quad$ | $\frac{dv}{dt}=x$ |
| particle $D$ | $\frac{dv}{dt}=-x$ |
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?
Getting Started
Remember that the differential of x means the 'rate of change' of $x$. The equation tells us exactly what that rate of change must be at each point.
What does a positive rate of change tell us about the changes in $x$? What does a negative rate of change tell us about the changes in $x$?
Student Solutions
Teachers' Resources
The key for this problem is for students to appreciate that the signs in differential equations are of crucial importance in determining the structure of a solution. They can make the difference between solutions growing to infinity, oscillating or settling down to zero.
When constructing differential equations using, for example, $F \; = \; ma$, negative signs on the right hand side correspond to 'repulsions' and positive signs correspond to 'attractions'.
Understanding the structure of equations in this way is a very powerful approach which can transcend the details of the algebraic solution.