Undetermined Coefficients Method for Scalar Equations

Given the equation:

We first find the complementary solution (see characteristic equation):

Then comes the hard part, finding the particular solution. We first need to look back at our original equation and do two things:

1. Compare it to our equations in our try functions.
   is of the 3rd form on the list.

2. Guess at a probable solution.
   Looking back at our original equation, , it seems be probable that our particular solution would be of the form as all derivatives are multiplies of .

Thus we plug our guess into our original equation to find that the particular solution yields .

We can now combine our complimentary and particular solution together and get our general solution:

Example from Elementary Differential Equations - Kohler & Johnson - Second Edition - Pg. 159

#incomplete #flag-review
Given the higher order differential equation of the form

One must first find the complementary-solution. Afterward you can use the following equation and Cramer's Rule to solve for and :

Now using the aforementioned Cramer's Rule we obtain:

Where is the Wronskian, and are the components of the complimentary solution, and is given in the initial problem.

Source: Diffeq Final Exam Review -- not linked.