Homogeneous Linear Equations With Constant Coefficients

Consider the nth-order linear equation with constant coefficients

displaymath119

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

(1)
Write the characteristic equation

displaymath120

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 tex2html_wrap_inline137 ;
(2.2)
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;

(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

displaymath155

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

displaymath121

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

displaymath171

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

displaymath175

Solution: Let us follow these steps:

(1)
Characteristic equation

displaymath177

Its roots are the complex numbers

displaymath179

In the analytical form, these roots are

displaymath181;

(2)
Independent set of solutions
(2.1)
The complex roots tex2html_wrap_inline183 and tex2html_wrap_inline185 generate the two solutions

displaymath187;

(2.2)
The complex roots tex2html_wrap_inline189 and tex2html_wrap_inline191 generate the two solutions

displaymath193;

(3)
The general solution is

displaymath195

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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Author: Mohamed Amine Khamsi

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