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.