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 functionis 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 bywhere 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 bywhere are arbitrary constants, are linearly independents solutions of the associated homogeneous equation (

*H*), and is a particular solution of (*NH*).

**
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