# Higher Order Linear Equations: Introduction and Basic Results

Let us consider the equation

,

and its associated homogeneous equation

The following basic results hold:

(1)
Superposition principle
Let 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 ;

(2)
The general solution of the equation (H) is given by

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;

(3)
The general solution of the equation (NH) is given by

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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