Homogeneous Linear Equations With Constant Coefficients

Consider the nth-order linear equation with constant coefficients


with tex2html_wrap_inline127 . In order to generate n linearly independent solutions, we need to perform the following:

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).

First case: Simple root
Let r be a simple root of the characteristic equation.
If r is a real number, then it generates the solution tex2html_wrap_inline137 ;
If tex2html_wrap_inline139 is a complex root, then since the coefficients of the characteristic equation are real, tex2html_wrap_inline141 is also a root. The two roots generate the two solutions tex2html_wrap_inline143 and tex2html_wrap_inline145;

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 tex2html_wrap_inline139 is a complex number, then tex2html_wrap_inline141 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 tex2html_wrap_inline167 . 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:

Characteristic equation


Its roots are the complex numbers


In the analytical form, these roots are


Independent set of solutions
The complex roots tex2html_wrap_inline183 and tex2html_wrap_inline185 generate the two solutions


The complex roots tex2html_wrap_inline189 and tex2html_wrap_inline191 generate the two solutions


The general solution is


As you may have noticed in this example, complex numbers do get involved very much in this kind of problem...

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Author: Mohamed Amine Khamsi

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