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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Homogeneous Equations with Constant Coefficients, Cont'd

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 p2 − 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