This method is interesting whenever the
previous method does not apply (when
*g*(*x*) is not of the desired form). The general idea is similar to
what we did for second order linear equations except that, in that
case, we were dealing with a small system and here we may be dealing
with a bigger one (depending on the order of the differential
equation). Let us describe the general case (constant coefficients or
not). Consider the equation

Suppose that a set of independent solutions of the associated homogeneous equation is known. Then a particular solution can be found as

where the functions can be obtained from the following system:

.

The determinant of this system is the Wronskian of , which is not zero. Cramer's formulas will give

,

where *W*(*x*) is the Wronskian and is the
determinant obtained from the Wronskian *W* by replacing the
-column in the vector column (0,0,..,0,1). Consequently,
a particular solution to the equation (*NH*) is given by

Note that when the order of the equation is not high, you may want to
solve the system using techniques other than Cramer's formulas.

**Example:** Find a particular solution of

**Solution:** Let us follow these steps:

**(1)**- Characteristic equation
Since , the roots of the characteristic equation are . Therefore, a set of independent solutions is ;

**(2)**- A particular solution is given by , where are solutions of the system
;

**(3)**- The resolution of the system gives
After integration we get

;

**(4)**- A particular solution is given by
Note that the constant 1 in may be dropped since it is the solution of the associated homogeneous equation.

**
**

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