Higher Order Linear Equations: Introduction and Basic Results

Let us consider the equation


and its associated homogeneous equation


The following basic results hold:

Superposition principle
Let tex2html_wrap_inline42 be solutions of the equation (H). Then, the function


is also solution of the equation (H). This solution is called a linear combination of the functions tex2html_wrap_inline48;

The general solution of the equation (H) is given by


where tex2html_wrap_inline52 are arbitrary constants and tex2html_wrap_inline54 are n solutions of the equation (H) such that,


In this case, we will say that tex2html_wrap_inline54 are linearly independent. The function tex2html_wrap_inline62 is called the Wronskian of tex2html_wrap_inline64 . We have


Therefore, tex2html_wrap_inline66 , for some tex2html_wrap_inline68, if and only if, tex2html_wrap_inline70 for every x;

The general solution of the equation (NH) is given by


where tex2html_wrap_inline52 are arbitrary constants, tex2html_wrap_inline54 are linearly independents solutions of the associated homogeneous equation (H), and tex2html_wrap_inline82 is a particular solution of (NH).

[Differential Equations] [First Order D.E.] [Second Order D.E.]
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Author: Mohamed Amine Khamsi

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