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 Depdendent Variable

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 Dependent Variable

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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 