A linear second order differential equations is written as

When *d*(*x*) = 0, the equation is called ** homogeneous**, otherwise it is
called ** nonhomogeneous**. To a nonhomogeneous equation

,

we associate the so called **associated homogeneous equation**

For the study of these equations we consider the explicit ones given by

where *p*(*x*) = *b*(*x*)/*a*(*x*), *q*(*x*) = *c*(*x*)/*a*(*x*) and
*g*(*x*) = *d*(*x*)/*a*(*x*). If *p*(*x*), *q*(*x*) and *g*(*x*) are defined and
continuous on the interval *I*, then the IVP

,

where and are arbitrary numbers, has a
unique solution defined on *I*.

**Main result:** *The general solution to
the equation ( NH) is given by
*

* , *

*
where
*

**(i)**- is the general solution to the homogeneous associated
equation (
*H*); **(ii)**- is a particular solution to the equation (
*NH*).

In conclusion, we deduce that in order to solve the
nonhomogeneous equation (*NH*), we need to

- Step 1: find the general solution to the homogeneous associated
equation (
*H*), say ; - Step 2: find a particular solution to the equation (
*NH*), say ; - Step 3: write down the general solution to (
*NH*) as

**
**

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