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 solutionsIf is a complex number, then is also a root with multiplicity

*m*. The two complex roots will generate 2*m*independent solutions

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

**
**

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