The power of the general solution is that it gives us the ability to
quickly find the solution x(t) and therefore predict the future
behavior of the system under any particular set of conditions. In
order to solve for the adjustable parameters there typically must be
one condition for each degree of freedom. For a mechanical
system, often these are given as the initial conditions, the
initial position and velocity of each particle in the system.
The procedure for finding the particular solution is (1) write each
condition in terms of the general solution, (2) solve the resulting
set of equations for the adjustable parameters, and (3) write down the
general solution while substituting the particular values found for
the adjustable parameters. As an example, consider the SHO under the
boundary conditions of initial position 
and initial velocity
at time t=0.
Step 1 -- In terms of the general solution (8), the position at t=0 is
Similarly, we may find the velocity at t=0 from the definition
,
Step 2 -- Next, solving (9-10) for the adjustable parameters, we have
Step 3 -- Finally, we substitute these results into the general solution (8) to find the particular solution,
