7.2 Nonhomogeneous linear equations

In this section, we examine how to solve nonhomogeneous differential equations. The terminology and methods are different from those we used for homogeneous equations, so let’s start by defining some new terms.

General solution to a nonhomogeneous linear equation

Consider the nonhomogeneous linear differential equation

The associated homogeneous equation

is called the complementary equation    . We will see that solving the complementary equation is an important step in solving a nonhomogeneous differential equation.

Proof

To prove $y(x)$ is the general solution, we must first show that it solves the differential equation and, second, that any solution to the differential equation can be written in that form. Substituting $y(x)$ into the differential equation, we have

So $y(x)$ is a solution.

Now, let $z(x)$ be any solution to $a_2\left(x\right)y\text{″}+a_1\left(x\right)y^{\prime }+a_0\left(x\right)y=r\left(x\right).$ Then

so $z(x)-y_p(x)$ is a solution to the complementary equation. But, $c_1{y}_1(x)+c_2{y}_2(x)$ is the general solution to the complementary equation, so there are constants $c_1$ and $c_2$ such that

Hence, we see that $z(x)=c_1{y}_1(x)+c_2{y}_2(x)+y_p(x\text{).}$

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In the preceding section, we learned how to solve homogeneous equations with constant coefficients. Therefore, for nonhomogeneous equations of the form $ay\text{″}+b{y}^{\prime }+cy=r(x),$ we already know how to solve the complementary equation, and the problem boils down to finding a particular solution for the nonhomogeneous equation. We now examine two techniques for this: the method of undetermined coefficients and the method of variation of parameters.