# Homogeneous Linear Equations With Constant Coefficients Consider the nth-order linear equation with constant coefficients with . In order to generate n linearly independent solutions, we need to perform the following:

(1)
Write the characteristic equation Then, look for the roots. These roots will be of two natures: simple or multiple. Let us show how they generate independent solutions of the equation(H).

(2)
First case: Simple root
Let r be a simple root of the characteristic equation.
(2.1)
If r is a real number, then it generates the solution ;
(2.2)
If is a complex root, then since the coefficients of the characteristic equation are real, is also a root. The two roots generate the two solutions and ;

(3)
Second case: Multiple root
Let r be a root of the characteristic equation with multiplicity m. If r is a real number, then generate the m independent solutions If is a complex number, then is also a root with multiplicity m. The two complex roots will generate 2m independent solutions Using properties of roots of polynomial equations, we will generate n independent solutions . Hence, the general solution of the equation (H) is given by Therefore, the real problem in solving (H) has to do more with finding roots of polynomial equations. We urge students to practice on this.

Example: Find the general solution of Solution: Let us follow these steps:

(1)
Characteristic equation Its roots are the complex numbers In the analytical form, these roots are ;

(2)
Independent set of solutions
(2.1)
The complex roots and generate the two solutions ;

(2.2)
The complex roots and generate the two solutions ;

(3)
The general solution is As you may have noticed in this example, complex numbers do get involved very much in this kind of problem... [Differential Equations] [First Order D.E.] [Second Order D.E.]
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