# 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). [Differential Equations] [First Order D.E.] [Second Order D.E.]
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