If we have a homogeneous differential equation, and we find that the equation has solutions
and 
then
is also a solution. For example, suppose we start with a second order homogeneous differential equation
where
are in general functions of
and supposeand
satisfy this equation, so that
and
Multiplying these by
and
respectively and adding gives
so thatis also a solution.
A problem may also be broken down into more than one problem and each solved separately. The solutions to each problem can be added to give the solution to the full problem. For example,
may be broken down into the two problemswith solution
and with any solution
Then
is also a solution.
Often we want two express a solution to a problem such asin terms of certain elementary functions eg
First we expressin term of those elementary functions so that
then find the response of the system governed by
to each component
ofand we can add these responses to obtain the solution.
We can also write the solution to initial condition boundary value problems as the sum of solutions to two simpler problems in the same way. For example, we can write
(1)
as two simpler problems
and
The solution to (1) is the sum of the solutions to the simpler problems.