We discuss theory related to nonhomogeneous linear equations.

### Nonhomogeneous Linear Equations

We’ll now consider the nonhomogeneous linear second order equation

where the forcing function isn’t identically zero. The next theorem, an extension of Theorem thmtype:5.1.1, gives sufficient conditions for existence and uniqueness of solutions of initial value problems for (eq:5.3.1). We omit the proof, which is beyond the scope of this book.To find the general solution of (eq:5.3.1) on an interval where , , and are continuous, it’s necessary to find the general solution of the associated homogeneous equation

on . We call (eq:5.3.2) the*complementary equation*for (eq:5.3.1).

The next theorem shows how to find the general solution of (eq:5.3.1) if we know one
solution of (eq:5.3.1) and a fundamental set of solutions of (eq:5.3.2). We call a *particular solution*
of (eq:5.3.1); it can be any solution that we can find, one way or another.

- Proof
- We first show that in (eq:5.3.5) is a solution of (eq:5.3.3) for any choice of the
constants and . Differentiating (eq:5.3.5) twice yields
so
Now we’ll show that every solution of (eq:5.3.3) has the form (eq:5.3.5) for some choice of the constants and . Suppose is a solution of (eq:5.3.3). We’ll show that is a solution of (eq:5.3.4), and therefore of the form , which implies (eq:5.3.5). To see this, we compute

We say that (eq:5.3.5) is the *general solution of on *.

If , , and are continuous and has no zeros on , then Theorem thmtype:5.3.2 implies that the general solution of

on is , where is a particular solution of (eq:5.3.6) on and is a fundamental set of solutions of on . To see this, we rewrite (eq:5.3.6) as and apply Theorem thmtype:5.3.2 with , , and .To avoid awkward wording in examples and exercises, we won’t specify the interval when we ask for the general solution of a specific linear second order equation, or for a fundamental set of solutions of a homogeneous linear second order equation. Let’s agree that this always means that we want the general solution (or a fundamental set of solutions, as the case may be) on every open interval on which , , and are continuous if the equation is of the form (eq:5.3.3), or on which , , , and are continuous and has no zeros, if the equation is of the form (eq:5.3.6). We leave it to you to identify these intervals in specific examples and exercises.

For completeness, we point out that if , , , and are all continuous on an open interval , but does have a zero in , then (eq:5.3.6) may fail to have a general solution on in the sense just defined.

In this section we to limit ourselves to applications of Theorem thmtype:5.3.2 where we can guess at the form of the particular solution.

item:5.3.1b Imposing the initial condition in (eq:5.3.9) yields , so . Differentiating (eq:5.3.9) yields Imposing the initial condition here yields , so the solution of (eq:5.3.8) is The figure below shows a graph of this solution.

item:5.3.2b Imposing the initial condition in (eq:5.3.12) yields , so . Differentiating (eq:5.3.12) yields and imposing the initial condition here yields , so . Therefore the solution of (eq:5.3.11) is The figure below shows a graph of this solution.

#### The Principle of Superposition

The next theorem enables us to break a nonhomogeous equation into simpler parts, find a particular solution for each part, and then combine their solutions to obtain a particular solution of the original problem.

- Proof
- If then

It’s easy to generalize Theorem thmtype:5.3.3 to the equation

where thus, if is a particular solution of on for , , …, , then is a particular solution of (eq:5.3.14) on . Moreover, by a proof similar to the proof of Theorem thmtype:5.3.3 we can formulate the principle of superposition in terms of a linear equation written in the form that is, if is a particular solution of on and is a particular solution of on , then is a solution of on .### Text Source

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