Let us consider the equation
and its associated homogeneous equation
The following basic results hold:
is also solution of the equation (H). This solution is called a linear combination of the functions ;
where are arbitrary constants and are n solutions of the equation (H) such that,
In this case, we will say that are linearly independent. The function is called the Wronskian of . We have
Therefore, , for some , if and only if, for every x;
where are arbitrary constants, are linearly independents solutions of the associated homogeneous equation (H), and is a particular solution of (NH).
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Author: Mohamed Amine Khamsi