However, this is rarely the form in which one wants a
solution of (17.3). One would prefer solutions that

are **real-valued functions of** x rather that complex-valued functions of x. But
these can be had as well,

since if z = x + iy is a complex number, then

are both real numbers. Applying the Superposition Principle, it is easy to see that if

and

are two complex-valued solutions of (17.3), then

and

are both real-valued solutions of (17.3).

Let us now compute the series expansion of

and

The expression on the right hand side is readily
identified as the Taylor series expansion of cos(x). We thus

conclude

Similarly, one can show that

On the other hand, if one adds (17.33) to i times (17.34) one gets

or

Thus, the real part of is cos(x), while the pure
imaginary part of is sin(x).

We now have a means of interpreting the function

in terms of elementary functions (rather than as a power series); namely,

Thus,

I now want to show how (17.33) and (17.34) allow us to
write down the general solution of a differential

equation of the form

as a linear combination of real-valued functions.

Now when p^{2} − 4q < 0, then

are the (complex) roots of the characteristic equation

corresponding to (17.40) and

are two (complex-valued) solutions of (17.40). But since
(17.40) is linear, since and
are solutions so

are

and

Note that and
are both **real-valued functions.**

We conclude that if the characteristic equation corresponding to

has two complex roots

then the general solution is

Example 17.1. The differential equation

has as its characteristic equation

The roots of the characteristic equation are given by

These are distinct real roots, so the general solution is

Example 17.2. The differential equation

has

as its characteristic equation. The roots of the characteristic equation are given by

Thus we have a double root and the general solution is

Example 17.3. The differential equation

has

as its characteristic equation. The roots of the characteristic equation are

and so the general solution is