. LINEAR DIFFERENTIAL EQUATIONS

For equations with one main condition

(Those linear), you have permission

To take your solutions,

With firm resolutions,

And add them in superposition. *

Let’s say a little more about the solution in eq. (3.2). If a is negative, then let’s define a = -cu², where cu is a real number. The solution now becomes x(t) = Ae^iojt + Be-^iojt. Using e^ie = cos 6 + i sin 0, this can be written in terms of trig functions, if desired. Various ways of writing the solution are:

x(t) = Ae^' + Be-™'

x(t) = C cos cut+ D sin cut,

x(t) = Ecos(uut + 4>₁),

x(t) = Fsin(cut + 4>₂). (3.3)

The various constants here are related to each other. For example, C = E cos f₁ and D = -E sin f₁, which follow from the cosine sum formula. Note that there are two free parameters in each of the above expressions for x(t). These parameters are determined from the initial conditions (say, the position and speed at t = 0). Depending on the specifics of a given problem, one of the above forms will work better than the others.

If a is positive, then let’s define a = cu², where cu is a real number. The solution in eq. (3.2) now becomes x(t) = Ae^ut + Be-"*. Using e^e = cosh(9 + sinh<9,>

x(t) = Ae^ + Be-^

x(t) = C cosh cut+ Dsinh cut,

x(t) = Ecosh(cut + 4>₁),

x(t) = Fsinh(cut + 4>₂). (3.4)

Again, the various constants are related to each other. If you are unfamiliar with the hyperbolic trig functions, a few facts are listed in Appendix A.

Remarks: Although the solution in eq. (3.2) is completely correct for both signs of a, it is generally more illuminating to write the negative-a solutions in either the trig forms or the e^±iujt exponential form where the i’s are explicit.

As in the first example above, you may be concerned that although we have found two solutions to the equation, we might have missed others. But the general theory of differential equations says that our second-order linear equation has only two independent solutions. Therefore, having found two independent solutions, we know that we’ve found them all. *

The usefulness of this method of guessing exponential solutions cannot be overempha­sized. It may seem somewhat restrictive, but it works. The examples in the remainder of this chapter should convince you of this.

This is our method, essential,

For equations we solve, differential.

It gets the job done,

And it’s even quite fun.

We just try a routine exponential.