We discuss the solution of an th order nonhomogeneous linear differential equation, making use of variation of parameters to find a particular solution.

### Variation of Parameters for Higher Order Equations

#### Derivation of the method

We assume throughout this section that the nonhomogeneous linear equation

is normal on an interval . We’ll abbreviate this equation as , where When we speak of solutions of this equation and its complementary equation , we mean solutions on . We’ll show how to use the method of variation of parameters to find a particular solution of , provided that we know a fundamental set of solutions of .We seek a particular solution of in the form

where is a known fundamental set of solutions of the complementary equation and are functions to be determined. We begin by imposing the following conditions on : These conditions lead to simple formulas for the first derivatives of : These formulas are easy to remember, since they look as though we obtained them by differentiating (eq:9.4.2) times while treating as constants. To see that (eq:9.4.3) implies (eq:9.4.4), we first differentiate (eq:9.4.2) to obtain which reduces to because of the first equation in (eq:9.4.3). Differentiating this yields which reduces to because of the second equation in (eq:9.4.3). Continuing in this way yields (eq:9.4.4).The last equation in (eq:9.4.4) is Differentiating this yields Substituting this and (eq:9.4.4) into (eq:9.4.1) yields Since , this reduces to Combining this equation with (eq:9.4.3) shows that is a solution of (eq:9.4.1) if which can be written in matrix form as

The determinant of this system is the Wronskian of the fundamental set of solutions , which has no zeros on , by Theorem thmtype:9.1.4. Solving (eq:9.4.5) by Cramer’s rule yields where is the Wronskian of the set of functions obtained by deleting from and keeping the remaining functions in the same order. Equivalently, is the determinant obtained by deleting the last row and -th column of .Having obtained , we can integrate to obtain . As in Trench 5.7, we take the constants of integration to be zero, and we drop any linear combination of that may appear in .

#### Third Order Equations

If , then Therefore and (eq:9.4.6) becomes

#### Fourth Order Equations

If , then Therefore

and (eq:9.4.6) becomes

### Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)