A solution is an explicit mathematical formula for the values of the degrees of freedom as a function of time which satisfies the equation of motion at all times t. To be a general solution, the solution most also contain a set of adjustable parameters (unspecified constants which may take any value) equal in number to the sum of the orders of the highest derivatives appearing for each degree of freedom in the equation of motion.
In the example of the SHO, an example of a solution is
To verify this as a solution, we substitute it into the equation
of motion (5). To do this, we first note that
and 
. Thus, we have
for (5),
The above equation will hold for all times t provided that 
, or equivalently,
Thus (6) qualifies as a solution, but only if 
takes the value given in (7).
Note that because the equation is satisfied for only one value of
, does not count as an adjustable parameter. On
the other hand, we do have a solution for all values of B. Thus the
solution (6) has one adjustable parameter. To be a
general solution, however, we here require two adjustable
parameters because the only degree of freedom is x and the second
derivative of x appears in (5).
A solution to (5) which does have two adjustable parameters, and therefore is a general solution, is
where (7) defines and B and C are
adjustable parameters. We leave the verification of this as a
solution as a recommended exercise for the student.
