## Linearity and Superposition

If we have a homogeneous differential equation, and we find that the equation has solutionsand thenis also a solution. For example, suppose we start with a second order homogeneous differential equation

whereare in general functions ofand supposeand satisfy this equation, so that

and

Multiplying these byandrespectively 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 solutionand with any solutionThenis also a solution.

Often we want two express a solution to a problem such asin terms of certain elementary functions egFirst we expressin term of those elementary functions so thatthen find the response of the system governed byto each componentofand 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.