Particular Solutions for Linear Systems

The particular solution () to a linear system of differential equation is found by two different methods depending on the system itself. Given a differential equation of the form:

Where is a matrix. To solve for the particular solution we take the following steps:

1. Find the complementary solution to the linear system.
2. Decide our solving method:
   1. Observe matrix , is it a functional matrix? If so then we must use variation of parameters method else we can continue.
   2. Look at , and compare to our undetermined coefficients method try functions. If there isn't a matching function we have to use variation of parameters method if there is we can continue with undetermined coefficients method.